# Multivariable Calculus

## Vector Calculus

### Vector Differentiation

Solution If $A=xyz{\vec i}+xz^{2}{\vec j}-y^{3}{\vec k},B=x^{3}{\vec i}-xyz{\vec j}+x^{2}z{\vec k}\,$ calculate ${\frac {\partial ^{2}{\vec A}}{\partial y^{2}}}\times {\frac {\partial ^{2}{\vec B}}{\partial x^{2}}}$ at the point (1,1,0).
Solution Find ${\frac {dr}{dt}},{\frac {d^{2}r}{dt^{2}}}$ when $r=3i-6t^{2}j+4tk$

Solution If $r=\sin ti+\cos tj+tk\,$ Find ${\frac {dr}{dt}},{\frac {d^{2}r}{dt^{2}}},\left|{\frac {dr}{dt}}\right\vert ,\left|{\frac {d^{2}r}{dt^{2}}}\right\vert$

Solution If $r=\cos nti+\sin ntj\,$ where n is constant and t varies, prove that $r\times ({\frac {dr}{dt}})=nk\,$ and $r\cdot ({\frac {dr}{dt}})=0\,$.

Solution If $r=e^{{nt}}a+e^{{-nt}}b\,$ where a,b are constant vectors, show that $({\frac {d^{2}r}{dt^{2}}})-n^{2}r=0\,$

Solution If $r=a\cos \omega t+b\sin \omega t\,$,Show that $r\times {\frac {dr}{dt}}=\omega a\times b\,$ and ${\frac {d^{2}r}{dt^{2}}}=-\omega ^{2}r\,$

Solution If $u=t^{2}i-tj+(2t+1)k\,$ and $v=(2t-3)i+j-tk\,$, Find ${\frac {d}{dt}}(u\cdot v)\,$ and ${\frac {d}{dt}}(u\times v)\,$ where t=1.

Solution If $a=\sin \theta i+\cos \theta j+\theta k,b=\cos \theta i-\sin \theta j-3k,c=2i+3j-k\,$.Find ${\frac {d}{d\theta }}[a\times (b\times c)]\,$ at $\theta =0\,$

Solution A particle moves along a curve whose parametric equations are $x=e^{{-t}},y=2\cos 3t,z=\sin 3t\,$.Find the velocity and acceleration at t=0.

Solution A particle moves along the curve $x=t^{3}+1,y=t^{2},z=2t+5\,$ where t is the time.Find the components of its velocity and acceleration at t=1 in the direction of $i+j+3k\,$.

Solution A particle moves so that its position vector is given by $r=\cos \omega ti+\sin \omega tj\,$ where $\omega \,$ is a constant.Show that i).The velocity of the particle is perpendicular to r ii).The acceleration is directed towards the origin and has magnitude proportional to the distance from the origin.

Solution Show that if a,b,c are constant vectors,then $r=at^{2}+bt+c\,$ is the path of a particle moving with constant acceleration.

SolutionIf $f=\cos xyi+(3xy-2x^{2})j-(3x+2y)k\,$,find the value of ${\frac {\partial f}{\partial x}},{\frac {\partial f}{\partial y}},{\frac {\partial ^{2}f}{\partial x^{2}}},{\frac {\partial ^{2}f}{\partial y^{2}}},{\frac {\partial ^{2}f}{\partial x\partial y}}\,$.

Solution If $f=(2x^{2}y-x^{4})i+(e^{{xy}}-y\sin x)j+x^{2}\cos yk\,$.Verify that ${\frac {\partial ^{2}f}{\partial x\partial y}}={\frac {\partial ^{2}f}{\partial y\partial x}}\,$.

Solution If $\phi (xyz)=xy^{2}z\,$ and $f=xzi-xyj+yz^{2}k\,$.Find ${\frac {\partial ^{3}(\phi f)}{\partial x^{2}\partial z}}\,$ at (2,-1,1).

### Vector Integration

Solution If $f(t)=(t-t^{2})i+2t^{3}j-3k\,$,Find i).$\int f(t)\,dt\,$ ii).$\int _{1}^{2}f(t)\,dt\,$

Solution If $f(t)=ti+(t^{2}-2t)j+(3t^{2}+3t^{3})k\,$,find $\int _{0}^{1}f(t)\,dt\,$.

Solution Evaluate $\int _{0}^{1}(e^{t}i+e^{{-2t}}j+tk)\,dt\,$

Solution If $r=ti-t^{2}j+(t-1)k\,$ and $s=2t^{2}i+6tk\,$,Evaluate $\int _{0}^{2}r\cdot s\,dt\,$ and $\int _{0}^{2}r\times s\,dt\,$.

Solution Evaluate $\int _{0}^{2}a\cdot b\times c\,dt\,$ where $a=ti-3j+2tk,b=i-2j+2k,c=3i+tj-k\,$

Solution Given that $r(t)=2i-j+2k\,$ when t=2,$r(t)=4i-2j+3k\,$ when t=3, Show that $\int _{2}^{3}[r\cdot {\frac {dr}{dt}}]\,dt=10\,$.

Solution Evaluate $\int _{1}^{2}r\times {\frac {d^{2}r}{dt^{2}}}\,dt\,$ where $r=2t^{2}i+tj-3t^{3}k\,$

Solution If $r(t)=5t^{2}i+tj-r^{3}k\,$,prove that $\int _{1}^{2}r\times {\frac {d^{2}r}{dt^{2}}}\,dt=-14i+75j-15k\,$

Solution Evaluate $\int a\cdot [r\times {\frac {d^{2}r}{dt^{2}}}]\,dt\,$

Solution Evaluate $\int _{1}^{2}[a\cdot (b\times c)+a\times (b\times c)]\,dt\,$ where $a=ti-3j+2tk,b=i-2j+2k,c=3i+tj-k\,$

Solution The acceleration of a moving particle at any time t is given by ${\frac {d^{2}r}{dt^{2}}}=12\cos 2ti-8\sin 2tj+16tk\,$.Find the velocity v and displacement r at any time t,if t=0,v=0 and r=0.

Solution Find the value of r satisfying the equation ${\frac {d^{2}r}{dt^{2}}}=6ti-24t^{2}j+4\sin tk\,$ given that $r=2i+j,{\frac {dr}{dt}}=-i-3k\,$ at t=0.

Solution If the acceleration of a particle at any time t greater than or equal to zero is given by $a=3\cos ti+4\sin tj+t^{2}k\,$ and the velocity v and displacement r are zero at t=0, then find v and r at any time t.

Solution Integrate ${\frac {d^{2}r}{dt^{2}}}=-n^{2}r\,$

SolutionIf $f(x,y,z)=x^{3}+y^{3}+z^{3}+3xyz\,$ then find $\nabla f\,$

SolutionIf $f(x,y,z)=3x^{2}y-y^{3}z^{2}\,$,find $\nabla f\,$ at the point (1,-2,-1).

Solution If $r=xi+yj+zk\,$ and $r=|r|=(x^{2}+y^{2}+z^{2})^{{{\frac {1}{2}}}}\,$, Prove that

i). $\nabla f(r)=f'(r)\nabla f\,$ ii). $\nabla r=({\frac {1}{r}})r\,$

Solution If $\phi (x,y,z)=(3r^{2}-4r^{{{\frac {1}{2}}}}+6r^{{-{\frac {1}{3}}}})\,$,Show that $\nabla \phi =2(3-r^{{-{\frac {3}{2}}}}-r^{{-{\frac {7}{3}}}})r\,$

Solution If $u=x+y+z,v=x^{2}+y^{2}+z^{2},w=xy+yz+zx\,$ Prove that ${\mathrm {grad}}u\cdot [\nabla v\times \nabla w]=0\,$

Solution Evaluate $\nabla e^{{r^{2}}}\,$ where $r^{2}=x^{2}+y^{2}+z^{2}\,$

Solution Show that $(a\cdot \nabla )\phi =a\cdot \nabla \phi \,$

Solution If $F=[y{\frac {\partial f}{\partial z}}-z{\frac {\partial f}{\partial y}}]i+[z{\frac {\partial f}{\partial x}}-x{\frac {\partial f}{\partial z}}]j+[x{\frac {\partial f}{\partial y}}-y{\frac {\partial f}{\partial x}}]k\,$,Prove that i).$F=r\times \nabla f\,$ ii).$F\cdot r=0\,$ iii).$F\cdot \nabla f=0\,$

SolutionIf $u=3x^{2}y,v=xz^{2}-2y\,$,find $(\nabla u)\cdot (\nabla v)\,$

SolutionFind $\nabla \phi ,|\nabla \phi |\,$ where $\phi (x,y,z)=(x^{2}+y^{2}+z^{2})e^{{-(x^{2}+y^{2}+z^{2})^{{{\frac {1}{2}}}}}}\,$

SolutionIf $f=x^{2}yi-2xzj+2yzk\,$, find i). $divf\,$. ii). Evaluate $div[(x^{2}-y^{2})i+2xyj+(y^{2}-2xy)k]\,$

Solution If $a_{1}i+a_{2}j+a_{3}k\,$,prove that $\nabla \cdot a=(\nabla a_{1})\cdot i+(\nabla a_{2})\cdot j+(\nabla a_{3})\cdot k\,$

SolutionIf $a=(x+3y)i+(y-3z)j+(x-2z)k\,$,find $(a\cdot \nabla )a\,$

Solution Evaluate $\nabla \cdot (a\times r)r^{n}\,$ where a is a constant vector.

Solution Find $\nabla \times f\,$ or curl F, where

i). $F=x^{2}yi-2xzj+2yzk\,$ ii). $F=(x^{2}-y^{2})i+2xyj+(y^{2}-2xy)k\,$

Solution Prove that $curlcurlF=0\,$ where $F=zi+xj+yk\,$

Solution If $V=e^{{xyz}}(i+j+k)\,$,find $curlV\,$

Solution If $r=xi+yj+zk\,$,prove that i).$divr=3\,$

ii). If $r=xi+yj+zk\,$ show that $curlr=0\,$

Solution IF $f=xy^{2}i+2x^{2}yzj-3yz^{2}k\,$,Find ${\mathrm {div}}f,{\mathrm {curl}}f\,$.What are their values at$(1,-1,1)\,$

Solution Find the ${\mathrm {curl}}\,$ of the vector $V=(x^{2}+yz)i+(y^{2}+zx)j+(z^{2}+xy)k\,$ at the point$(1,2,3)\,$

Solution If $f=(x+y+1)i+j+(-x-y)k\,$,prove that $f\cdot {\mathrm {curl}}f=0\,$

Solution a). Prove that vector $f=(x+3y)i+(y-3z)j+(x-2z)k\,$ is solenoidal.

b). Determine the constant 'a' so that the vector $f=(x+3y)i+(y-2z)j+(x+az)k\,$ is solenoidal.

Solution a). Show that the vector $f=(\sin y+z)i+(x\cos y-z)j+(x-y)k\,$ is irrational. b). Determine the constants 'a','b','c' so that the vector $f=(x+2y+az)i+(bx-3y-z)j+(4x+cy+2z)k\,$ is irrational.

Solution Prove that $\nabla \cdot (r^{3}r)=6r^{3}\,$

Solution Prove ${\mathrm {div}}[r\nabla r^{{-3}}]=3r^{{-4}}\,$ or $\nabla \cdot [r\nabla ({\frac {1}{r^{3}}})]={\frac {3}{r^{4}}}\,$

Solution If a is a constant vector,prove that ${\mathrm {curl}}{\frac {a\times r}{r^{3}}}=-{\frac {a}{r^{3}}}+{\frac {3r}{r^{5}}}(a\cdot r)\,$

Solution Show that $\nabla ^{2}({\frac {x}{r^{3}}})=0\,$

Solution Show that ${\mathrm {div}}{\mathrm {grad}}(r^{m})=m(m+1)r^{{m-2}}\,$

Solution Evaluate ${\mathrm {curl}}{\mathrm {grad}}(r^{m})\,$ where $r=|r|=|xi+yj+zk|\,$

Solution If $u=x^{2}-y^{2}+4z\,$,Show that $\nabla ^{{2}}u=0\,$

Solution Show that $u=ax^{2}+by^{2}+cz^{2}\,$ satisfies Laplace equation $\nabla ^{2}u=0\,$

Solution If f and g are two scalar functions,prove that ${\mathrm {div}}(f\nabla g)=f\nabla ^{2}g+\nabla f\times \nabla g\,$

Solution Show that $\nabla \cdot (\nabla \times r)=0\,$ if $\nabla \times V=0\,$

Solution Prove that $\nabla ^{2}({\frac {1}{r}})=0\,$,where $r^{2}=x^{2}+y^{2}+z^{2}\,$

Solution Evaluate $\nabla ^{2}({\frac {x}{r^{2}}})\,$

Solution Prove that $V\times {\mathrm {curl}}V={\frac {1}{2}}\nabla V^{2}-(V\cdot \nabla )V\,$

SolutionIf $v=v_{1}i+v_{2}j+v_{3}k\,$,prove that $\nabla \times v=\nabla v_{1}\times i+\nabla v_{2}\times j+\nabla v_{3}\times k\,$

Solution If r(P) be the vector from the origin O to a point P in the xy-plane,then show that the plane scalar field $u(P)=\log r\,$ satisfies the equation $\nabla ^{2}u=0\,$

Solution Prove that ${\mathrm {div}}(A\times r)=r\cdot {\mathrm {curl}}A\,$

Solution Prove that $\nabla \times (F\times r)=2F-(\nabla \cdot F)r+(r\cdot \nabla )F\,$

Solution If $u=e^{{2x}}+x^{2}z\,$ and $v=2z^{2}y-xy^{2}\,$,find ${\mathrm {grad}}(uv)\,$ at the point (1,0,2).

Solution Prove that ${\mathrm {curl}}[r\times (a\times r)]=3r\times a\,$,where a is a constant vector.

Solution Find the unit normal to the surface $z=x^{2}+y^{2}\,$ at the point (-1,-2,5).

Solution Find the directional derivative of $\phi =x^{2}yz+2xz^{2}\,$ at (1,-2,-1) in the direction of $2i-j-2k\,$

Solution Calculate the maximum rate of change and the corresponding direction for the function $\phi =x^{2}y^{3}z^{4}\,$ at the point $2i+3j-k\,$

Solution Find the equation of the tangent plane and normal to the surface $xyz=4\,$ at the point (1,2,2).

Solution Find the equation of the tangent line and normal plane to the curve of intersection of $x^{2}+y^{2}+z^{2}=1,x+y+z=1\,$ at (1,0,0).

Solution Find the angle between the curves $x^{2}+y^{2}+z^{2}=9,z=x^{2}+y^{2}-3\,$ at the point (2,-1,2).

Solution Find the constants a and b so that surfaces $ax^{2}-byz=(a+2)x\,$ will be orthogonal to the surface $4x^{2}y+z^{3}=4\,$ at the point (1,-1,2).

### Line, Surface & Volume Integrals

Solution If $F=3xyi-y^{2}j\,$,Evaluate $\int _{{C}}F\cdot dr\,\,$,where C is the curve $y=2x^{2}\,$ in the xy-plane from (0,0)to (1,2).

Solution Evaluate $\int _{{C}}F\cdot \,dr\,$ where $F=(x^{2}+y^{2})i-2xyj\,$,C is the rectangle in xy-plane bounded by $y=0,x=a,y=b,x=0\,$

Solution If $F=(2x+y)i+(3y-x)j\,$,evaluate $\int _{{C}}F\cdot \,dr\,$ where C is the curve in the xy-plane consisting of the straight line from $(0,0)\,$ to $(2,0)\,$ and then to $(3,2)\,$

Solution If $F=(3x^{2}+6y)i-14yzj+20xz^{2}k\,$,evaluate $\int _{{C}}F\cdot \,dr\,$ where C is the straight line joining $(0,0,0),(1,1,1)\,$

Solution Evaluate $\int _{{C}}F\cdot \,dr\,$ where $F=\cos yi-x\sin yj\,$ and C is the curve $y={\sqrt {1-x^{2}}}\,$ in the xy-plane from $(1,0)\,$ to $(0,1)\,$

Solution Evaluate $\int F\cdot \,dr\,$ along the curve $x^{2}+y^{2}=1,z=1\,$ in the positive direction from (0,1,1) to (1,0,1),where $F=(2x+yz)i+xzj+(xy+2z)k\,$

Solution Evaluate $\int _{{C}}F\cdot \,dr\,$ where $F=x^{2}y^{2}i+yj\,$ and the curve c is $y^{2}=4x\,$ in the xy-plane from (0,0) to (4,4) where $r=xi+yj\,$

Solution Evaluate $\int _{{C}}F\cdot \,dr\,$ where $F=xyi+(x^{2}+y^{2})j\,$ and C in the arc of the curve $y=x^{2}-4\,$ from (2,0) to (4,12).

Solution If $F=(2x^{2}+y^{2})i+(3y-4x)j\,$ evaluate $\int F\cdot \,dr\,$ around the triangle ABC whose vertices are $A(0,0),B(2,0),C(2,1)\,$

Solution If $F=(2y+3)i+xzj+(yz-x)k\,$.evaluate $\int _{{C}}F\cdot \,dr\,$ where C is the path consisting of the straight lines from (0,0,0) to (0,0,1) then to (0,1,1) and then to (2,1,1).

Solution If $A=(2y+3)i+xzj+(yz-x)k\,$,evaluate $\int _{{C}}A\cdot \,dr\,$ along the curve C.$x=2t^{2},y=t,z=t^{3}\,$ from t=0 to t=1.

Solution Evaluate $\int _{{C}}F\cdot \,dr\,$ where $F=zi+xj+yk\,$ and C is the arc of the curve $r=\cos ti+\sin tj+tk\,$ from $t=0\,$ to $t=\pi \,$

Solution Evaluate $\int _{{C}}F\cdot \,dr\,$ where $F=yzi+zxj+xyk\,$ and C is the portion of the curve $r=a\cos ti+b\sin tj+ctk\,$ from $t=0\,$ to $t={\frac {\pi }{2}}\,$

Solution Evaluate $\int _{{C}}F\cdot \,dr\,$ where $F=xyi+yzj+zxk\,$ and C is the arc of the curve $r=a\cos \theta i+a\sin \theta j+a\theta k\,$ from $\theta =0\,$ to $\theta ={\frac {\pi }{2}}\,$

Solution If $F=xyi-zj+x^{2}k\,$,evaluate $\int _{{C}}F\times \,dr\,$ where C is the curve $x=t^{2},y=2t,z=t^{3}\,$ from t=0 to 1.

Solution Find the total work done in moving a particle in a force field given by $F=3xyi-5zj+10xk\,$ along the curve $x=t^{2}+1,y=2t^{2},z=t^{3}\,$ from t=1 to t=2.

Solution Find the work done when a force $F=(x^{2}-y^{2}+x)i-(2xy+y)j\,$ moves a particle in xy-plane from (0,0) to (1,1) along the curve $y^{2}=x\,$

Solution Find the work done in moving a particle in a force field $F=3x^{2}i+(2xz-y)j+zk\,$ along the line joining (0,0,0) to (2,1,3).

Solution Find the work done in moving a particle once round a circle C in the xy-plane,if the circle has centre at the origin and radius 3 and when the force field is given by $F=(2x-y+z)i+(x+y-z^{2})j+(3x-2y+4z)k\,$

Solution Find the circulation of F round the curve C where $F=yi+zj+zxk\,$ and C is the circle $x^{2}+y^{2}=1,z=0\,$

Solution Find the circulation of F round the curve C where $F=e^{x}\sin yi+e^{x}\cos yj\,$ and C is the rectangle whose vertices are $(0,0),(1,0),(1,{\frac {\pi }{2}}),(0,{\frac {\pi }{2}})\,$

Solution Evaluate $\iint _{{S}}(y^{2}z^{2}i+z^{2}x^{2}j+x^{2}y^{2}k)\cdot \,ds\,$ where S is the part of the sphere $x^{2}+y^{2}+z^{2}=1\,$ above the xy-plane.

Solution If $F=yi+(x-2xz)j-xyk\,$,evaluate $\iint _{{S}}(\nabla \times F)\cdot n\,dS\,$ where S is the surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}\,$ above the xy-plane.

Solution Evaluate $\iint _{{S}}({\mathrm {curl}}F\cdot n\,dS\,$ where $F=yi+zj+xk\,$ and surface S in the part of the sphere $x^{2}+y^{2}+z^{2}=1\,$ above the xy-plane.

Solution Evaluate $\iint _{S}(y^{2}zi+z^{2}xj+x^{2}yk)\cdot dS\,$ where S is the surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}\,$ lying in the positive octant.

Solution Evaluate $\iint _{{S}}F\cdot n\,dS\,$ over the surface of the cylinder $x^{2}+y^{2}=9\,$ included in the first octant between z=0 and z=4 where $F=zi+xj-yzk\,$

Solution Evaluate $\iint _{{S}}F\cdot n\,dS\,$ where $F=2yxi-yzj+x^{2}k\,$ over the surface of the cube bounded by the coordinate planes and planes $x=a,y=a,z=a\,$

Solution Evaluate $\iint _{{S}}F\cdot n\,dS\,$ where $F=(x-z)i+(x^{3}+yz)j-3xy^{2}k\,$ and S in the surface of the cone $z=2-{\sqrt {(x^{2}+y^{2})}}\,$ above the xy-plane.

Solution Evaluate $\iiint _{V}F\,dv\,$ where $F=xi+yj+zk\,$ and V is the region bounded by the surfaces $x=0,x=2,y=0,y=6,z=4,z=x^{2}\,$

Solution Evaluate $\iiint _{V}\phi \,dv\,$ where $\phi =45x^{2}y\,$ and V is the closed region bounded by the planes $4x+2y+z=8,x=0,y=o,z=0\,$

Solution If $F=2xzi-xj+y^{2}k\,$ evaluate $\iiint _{V}F\,dV\,$ where V is the region bounded by the planes $x=y=z=0,x=y=z=1\,$

Solution Let r denote the position vector any point (x,y,z) measured from an origin O and let $r=|r|\,$.Evaluate $\iint _{S}{\frac {r}{|r|^{3}}}\cdot dS\,$ where S denotes the sphere of radius a with center at the origin.

Solution Evaluate $\iint _{S}F\cdot ndS\,$,where $F=yi+2xj-zk\,$ and S in the surface of the plane 2x+y=6 in the first octant cut off by the plane z=4.

Solution If $F=(2x^{2}-3z)i-2xyj-4xk\,$,then evaluate i). $\iiint _{V}\nabla \times FdV\,$ ii). $\iiint _{V}FdV\,$, where V is the region bounded by x=0,y=0,z=0 and $2x+2y+z=4\,$

Solution Evaluate $\iiint _{V}(2x+y)dV\,$ where V is closed region bounded by the cylinder $z=4-x^{2}\,$ and the planes x=0,y=0,y=2 and z=0.

Solution Find the volume of the region common to the intersecting cylinders $x^{2}+y^{2}=a^{2}\,$ and $x^{2}+z^{2}=a^{2}\,$

### Green, Stokes & Gauss Divergence Theorems

• Green's Theorem in the plane - Relation between plane and line integrals

If R is a closed region in the xy-plane bounded by a simple closed curve C and if $\phi (x,y)\,$ and $\psi (x,y)\,$ are continuous functions having continuous partial derivatives in R,then $\oint (\psi dx+\phi dy)=\iint _{R}\left[{\frac {\partial \phi }{\partial x}}-{\frac {\partial \psi }{\partial y}}\right]dxdy\,$ where C is traversed in the positive (anti-clockwise) direction.

• Stokes Theorem - Relation between surface and line integrals

If F is any continuously differentiable vector point function and S is a surface bounded by a curve C,then $\oint F\cdot dr=\iint _{S}{\mathrm {curl}}F\cdot ndS\,$ where the unit normal n at any point of S is drawn in the direction in which a right-handed screw would move when rotated in the sense of description of C.

Solution If $F=(x^{2}-y^{2})i+2xyj\,$ and $r=xi+yj\,$,find the value of $\int F\cdot dr\,$ around the rectangular boundary x=0,x=a,y=0,y=b.

SolutionEvaluate by Green's theorm in plane$\int _{C}(e^{{-x}}\sin ydx+e^{{-x}}\cos ydy)\,$ where C is the rectangle with vertices (0,0),$(\pi ,0),(\pi ,{\frac {\pi }{2}}),(0,{\frac {\pi }{2}})\,$.

Solution Verify Green's theorm in plane for $\int _{C}[(x^{2}-xy^{3})dx+(y^{2}-2xy)dy]\,$ where C is the square with vertices (0,0),(2,0),(2,2),(0,2).

Solution Verify Green's theorm in plane for $\oint _{C}[(3x^{2}-8y^{2})dx+(4y-6xy)dy]\,$ where C is the boundary of the region defined by x=0,y=0,$x+y=1\,$

Solution Apply Green's theorm in the plane to evaluate $\int _{C}[(2x^{2}-y^{2})dx+(x^{2}+y^{2})dy]\,$ where C is the boundary of the curve enclosed by the x-axis and the semi-circle$y=(1-x^{2})^{{{\frac {1}{2}}}}\,$

Solution Show that the area bounded by a simple closed curve C is given by ${\frac {1}{2}}\int _{C}(xdy-ydx)\,$. Hence deduce that the area of the ellipse ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1\,$

Solution Verify Green's theorm in a plane $\oint _{C}[(x^{2}-2xy)dx+(x^{2}y+3)dy]\,$ where C is the boundary of the region defined by $y^{2}=8x,x=2\,$

Solution Evaluate by Green's theorm $\oint _{C}[(\cos x\sin y-xy)dx+\sin x\cos ydy]\,$ where C is the circle $x^{2}+y^{2}=1\,$

Solution Evaluate by Green's theorm in the plane $\oint _{C}[(x^{2}-\cos hy)dx+(y+\sin x)dy]\,$ where C is the rectangle with vertices (0,0),($\pi \,$,0),($\pi \,$,1),(0,1).

Solution Evaluate $\oint _{C}[(y-\sin x)dx+\cos xdy]\,$ where C is the triangle whose vertices are (0,0),(${\frac {\pi }{2}}\,$,0),(${\frac {\pi }{2}}\,$, by using Green's theorm in plane.

Solution Verify Green's theorm in the plane for $\oint _{C}[(xy+y^{2})dx+x^{2}dy]\,$ where C is the closed curve of the region bounded by $y=x,y=x^{2}\,$

Solution Verify Green's theorm in plane for $\oint _{C}[(3x^{2}-8y^{2})dx+(4y-6xy)dy]\,$ where C is the region bounded by the parabolas $y^{2}=x,y=x^{2}\,$

Solution Verify Stokes' theorm for $F=(x^{2}+y^{2})i-2xyj\,$ taken round the rectangle bounded by $x=\pm a,y=0,y=b\,$

Solution Evaluate $\oint _{C}F\cdot dr\,$ by Stokes' theorm where $F=y^{2}i+x^{2}j-(x+z)k\,$ and C is the boundary of the triangle with vertices at (0,0,0),(1,0,0),(1,1,0).

Solution If $F=(2x^{2}+y^{2})i+(3y-4x)j\,$ evaluate $\oint _{C}F\cdot dr\,$ where C is the boundary of the triangle with vertices (0,0),(2,0),(2,1)

Solution Verify Stokes'theorm for the function $F=x^{2}i+xyj\,$ integrated along the rectangle in the plane z=0,where sides are along the lines x=0,y=0,x=a and y=b.

Solution Evaluate by Stokes'theorm $\oint _{C}(e^{x}dx+2ydy-dz)\,$ where C is the curve $x^{2}+y^{2}=4,z=2\,$

Solution Evaluate by Stokes'theorm $\oint _{C}(\sin zdx-\cos xdy+\sin ydz)\,$ where C is the boundary of the rectangle $0\leq x\leq \pi ,0\leq y\leq 1,z=3\,$

Solution By converting into line integral,evaluate $\iint _{S}(\nabla \times A)\cdot ndS\,$, where $A=(x-z)i+(x^{3}+yz)j-3xy^{2}k\,$ and S is the surface of the cone $z=2-{\sqrt {(x^{2}+y^{2})}}\,$ above the xy-plane.

Solution By converting into a line integral evaluate $\iint _{S}(\nabla \times F)\cdot ndS\,$ where $F=(x^{2}+y-4)i+3xyj+(2xy+z^{2})k\,$ and S is the surface of the paraboloid $z=4-(x^{2}+y^{2})\,$ above the xy-plane.

Solution Evaluate $\iint _{S}(\nabla \times F)\cdot ndS\,$ where $F=(y-z+2)i+(yz+4)j-xzk\,$ and S is the surface of the cube $x=y=z=0,x=y=z=2\,$ above the xy-plane.

Solution Verify Stokes'theorm for $F=yi+zj+xk\,$ where S is the upper half surface of the sphere $x^{2}+y^{2}+z^{2}=1\,$ and C is its boundary.

Solution Verify Stokes'theorm for the vector $F=3yi-xzj+yz^{2}k\,$ where S is the surface of the paraboloid $2z=x^{2}+y^{2}\,$ bounded by z=2 and C is its boundary.

Solution Verify Stokes'theorm for the function $F=zi+xj+yk\,$ where C is the unit circle in xy-plane bounding the hemisphere $z={\sqrt {(1-x^{2}-y^{2})}}\,$

Solution Apply Stokes'theorm to prove that $\int _{C}(ydx+zdy+xdz)=-2{\sqrt {2}}\pi a^{2}\,$,where C is the curve given by $x^{2}+y^{2}+z^{2}-2ax-2ay=0,x+y=2a\,$ and begins at the point (2a,0,0) and goes at first below the xy-plane.

Solution By Stokes' theorem,prove that ${\mathrm {curl}}{\mathrm {grad}}\phi =0\,$

Solution Evaluate $\iint _{S}F\cdot ndS\,$ where $F=axi+byj+czk\,$ and S is the surface of the sphere $x^{2}+y^{2}+z^{2}=1\,$

Solution Use divergence theorm to find $\iint _{S}F\cdot ndS\,$ for the vector $F=xi-yj+2zk\,$ over the sphere $x^{2}+y^{2}+(z-1)^{2}=1\,$

Solution If $F=axi+byj+czk\,$ where a,b,c are constants,show that $\iint _{S}(n\cdot F)dS={\frac {4}{3}}\pi (a+b+c)\,$,S being the surface of the sphere $(x-1)^{2}+(y-2)^{2}+(z-3)^{2}=1\,$

Solution Find $\iint _{S}A\cdot ndS\,$,where $A=(2x+3z)i-(xz+y)j+(y^{2}+2z)k\,$ and S is the surface of the sphere having center at (3,-1,2) and radius 3 units.

Solution By using the Gauss Divergence theorm,evaluate $\iint _{S}(xdydz+ydzdx+zdxdy)\,$,where S is the surface of the sphere $x^{2}+y^{2}+z^{2}=4\,$

Solution Apply divergence theorm to evaluate $\iint _{S}[(x+z)dydz+(y+z)dzdx+(x+y)dxdy]\,$ where S is the surface of the sphere $x^{2}+y^{2}+z^{2}=4\,$

Solution If $F=4xzi-y^{2}j+yzk\,$,then evaluate $\iint _{s}F\cdot ndS\,$ where S is the surface of the cube enclosed by x=0,x=1,y=0,y=1,z=0 and z=1.

Solution Evaluate $\iint F\cdot ndS\,$,where $F=4xyi+yzj-xzk\,$ and S is the surface of the cube bounded by the planes x=0,x=2,y=0,y=2,z=0,z=2.

Solution Apply Gauss'theorm to evaluate $\iint _{S}[(x^{3}-yz)dzdx-2x^{2}ydzdx+zdxdy]\,$ over the surface S of a cube bounded by the coordinate planes and the planes x=y=z=a.

Solution Apply Gauss'theorm to show that $\iint _{S}(x^{3}-yz)i-2x^{2}yj+2k]\cdot ndS={\frac {a^{5}}{3}}\,$,where S denotes the surface of the cube bounded by the planes $x=0,x=a,y=0,y=a,z=0,z=a\,$

Solution Evaluate $\iint _{s}[x^{2}dydz+y^{2}dzdx+2z(xy-x-y)dxdy]\,$ where S is the surface of the cube $0\leq x\leq 1,0\leq y\leq 1,0\leq z\leq 1\,$

Solution Find the value of $\iint _{S}(F\times \nabla \phi )\cdot ndS\,$ where $F=x^{2}i+y^{2}j+z^{2}k,\phi =xy+yz+zx,S:x=\pm 1,y=\pm 1,z=\pm 1\,$

Solution Evaluate $\iint _{S}F\cdot ndS\,$,where $F=xi-yj+(z^{2}-1)k\,$ and S is the closed surface bounded by the planes $z=0,z=1\,$ and the cylinder $x^{2}+y^{2}=4\,$ by the application of Gauss'theorm.

Solution Use Gauss' theorem to evaluate the integral $\iint _{S}F\cdot ndS\,$ of the vector field $F=xy^{2}i+y^{3}j+y^{2}zk\,$ through the closed surface formed by the cylinder $x^{2}+y^{2}=9\,$ and the plane $z=0,z=2\,$

Solution Use Gauss divergence theorem to find $\iint _{S}F\cdot ndS\,$ where $F=2x^{2}yi-y^{2}j+4xz^{2}k\,$ and S is the closed surface in the first octant bounded by $y^{2}+z^{2}=9,x=2\,$

Solution Evaluate $\iint _{S}(zx^{2}dxdy+x^{3}dydz+yx^{2}dzdx)\,$ where S is the closed surface consisting of the cylinder $x^{2}+y^{2}=4\,$ and the circular discs $z=0,z=3\,$

Solution If $F=\nabla \phi ,\nabla ^{2}\phi =-4\pi \rho \,$ show that $\iint _{S}F\cdot ndS=-4\pi \iiint _{V}\rho dV\,$

SolutionIf $\phi \,$ is harmonic in V,then $\iint _{S}{\frac {\partial \phi }{\partial n}}dS=\iiint _{V}\nabla ^{2}\phi dV\,$ where S is the surface enclosing V.

Solution Prove that i). $\iiint _{V}\nabla \phi \cdot AdV=\iint _{S}\phi \cdot ndS-\iiint _{V}\phi \nabla \cdot AdV\,$ ii).Prove that $\iiint _{V}F\cdot {\mathrm {curl}}GdV=\iint _{S}G\times F\cdot dS+\iiint _{V}G\cdot {\mathrm {curl}}FdV\,$

Solution Show that for any closed surface S, i). $\iint _{S}ndS=0\,$ ii).$\iint _{S}r\times ndS=0\,$ iii).$\iint _{S}(\nabla \phi )\times ndS=0\,$

Solution If V is the volume of a region T bounded by a surface S,then prove that $V=\iint _{S}xdydz=\iint _{S}ydzdx=\iint _{S}zdxdy\,$

Solution Evaluate $\iint _{S}(\nabla \times F)\cdot ndS\,$,where $F=(x-z)i+(x^{3}+yz)j-3xy^{2}k\,$ and S is the surface of the cone $z=2-{\sqrt {(x^{2}+y^{2})}}\,$ above the xy plane.

Solution Evaluate $\iint _{S}(x^{3}dydz+y^{3}dzdx+z^{3}dxdy)\,$ by converting the surface integral into a volume integral.Here,S is the surface of the sphere $x^{2}+y^{2}+z^{2}=1\,$

Solution Evaluate with the help of divergence theorm the integral $\iint _{S}[xz^{2}dydz+(x^{2}y-z^{3})dzdx+(2xy+y^{2}z)dxdy]\,$, where S is the entire surface of the hemispherical region bounded by $z={\sqrt {(a^{2}-x^{2}-y^{2})}},z=0\,$

Solution Evaluate $\iint _{S}(ax^{2}+by^{2}+cz^{2})dS\,$ over the sphere $x^{2}+y^{2}+z^{2}=1\,$ using the divergence theorm.

Solution Compute $\iint _{S}(a^{2}x^{2}+b^{2}y^{2}+c^{2}z^{2})^{{{\frac {1}{2}}}}dS\,$ over the ellipsoid $ax^{2}+by^{2}+cz^{2}=1\,$

Solution Compute $\iint _{S}(a^{2}x^{2}+b^{2}y^{2}+c^{2}z^{2})^{{{\frac {-1}{2}}}}dS\,$ over the ellipsoid $ax^{2}+by^{2}+cz^{2}=1\,$

Solution Evaluate$\iint _{S}F\cdot ndS\,$ over the entire surface of the region above the xy plane bounded by the cone $z^{2}=x^{2}+y^{2}\,$ and the plane z=4,if $F=xi+yj+z^{2}k\,$

Solution By using the Gauss divergence theorm evaluate $\iint _{S}(xi+yj+z^{2}k)\cdot ndS\,$,where S is the closed surface bounded by the cone $x^{2}+y^{2}=z^{2}\,$ and the plane z=1.

Solution Evaluate $\iint _{S}(\nabla \times F)\cdot ndS\,$ where $A=[xye^{z}+\log(z+1)-\sin x]k\,$ and S is the surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}\,$ above the xy plane.

Solution Evaluate $\iint _{S}xyzdS\,$ where S is the surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}\,$

Solution By transforming to a triple integral,evaluate $\iint _{S}(x^{3}dydz+x^{2}ydzdx+x^{2}zdxdy)\,$ where S is the closed surface bounded by the planes $z=0,z=b\,$ and the cylinder $x^{2}+y^{2}=a^{2}\,$

Solution Evaluate $\iint _{S}{\frac {1}{p}}dS\,$ where S is the surface of the ellipsoid ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1\,$ and p is the perpendicular drawn from the origin to the tangent plane at $(x,y,z)\,$

Solution Show that $\iint _{S}(x^{2}i+y^{2}j+z^{2}k)\cdot ndS\,$ vanishes where S denotes the surface of the ellipsoid ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1\,$

Solution Verify the divergence theorm theorm for $F=4xzi-y^{2}j+yzk\,$ taken over the cube bounded by $x=0,x=1,y=0,y=1,z=0,z=1\,$

Solution Evaluate $\iint _{S}(x^{3}dydz+y^{3}dzdx)\,$ where S is the surface of hte sphere $x^{2}+y^{2}+z^{2}=a^{2}\,$

Solution Show that the vector field $F=(2xy^{2}+yz)i+(2x^{2}y+xz+2yz^{2})j+(2y^{2}z+xy)k\,$ is conservative.

Solution Show that the vector field defined by $F=(2xy-z^{3})i+(x^{2}+z)j+(y-3xz^{2})k\,$ is conservative and find the scalar potential of F.

Solution Show that the vector field F given by $F=(y+\sin z)i+xj+x\cos zk\,$ is conservative,find its scalar potential.

Solution Show that $F=xi+yj+zk\,$ is conservative and find $\phi \,$ such that $F=\nabla \phi \,$

Solution Prove that $F=|r|^{2}r\,$ is conservative and find its scalar potential.

Solution Show that $(y^{2}z^{3}\cos x-4x^{3}z)dx+2z^{3}y\sin xdy+(3y^{2}z^{2}\sin x-x^{4})dz\,$ is an exact differential of some function $\phi \,$ and find this function.

Solution Show that $(2x\cos y+z\sin y)dx+(xz\cos y-x^{2}\sin y)dy+x\sin ydz=0\,$ is an exact differential and hence solve it.

Solution Evaluate $\int _{C}[2xyz^{2}dx+(x^{2}z^{2}+z\cos yz)dy+(2x^{2}yz+y\cos yz)dz\,$ where C is any path from $(0,0,0)\,$ to $1,{\frac {\pi }{4}},2)\,$

Solution If $F=\cos yi-x\sin yj\,$, evaluate $\int _{C}F\cdot dr\,$ where C is the curve $y={\sqrt {(1-x^{2})}}\,$ in the xy plane from $(1,0)\,$ to $(0,1)\,$

Solution Evaluate $\int _{C}[yzdx+(xz+1)dy+xydz]\,$,where C is any path from $(1,0,0)\,$ to $(2,1,4)\,$

Solution A vector field is given by $F=(x^{2}+xy^{2})i+(y^{2}+x^{2}y)j\,$ Show that the field is irrational and obtain its scalar potential.

Solution Show that the vector field F given by $F=(x^{2}-yz)i+(y^{2}-zx)j+(z^{2}-xy)k\,$ is irrotational. Find a scalar $\phi \,$ such that $F=\nabla \phi \,$

Solution Show that the vector function $F=(\sin y+z\cos x)i+(x\cos y+\sin z)j+(y\cos z+\sin x)k\,$ is irrotational and find the scalar function $\phi \,$ such that $F=\nabla \phi \,$

## Multiple Integrals

solution $\int _{0}^{2}\int _{0}^{1}(2x+y)^{8}dxdy\,$

solution Evaluate $\int _{0}^{2}\int _{1}^{2}(x^{2}+y^{2})dxdy\,$

solution Evaluate $\int _{0}^{1}\int _{1}^{2}(x^{2}+y^{2})dxdy\,$

solution $\int _{0}^{3}\int _{1}^{2}xy(x+y)dxdy\,$

solution $\int _{0}^{a}\int _{0}^{b}(x^{2}+y^{2})dxdy\,$

solution $\int _{1}^{2}\int _{3}^{4}{\frac {1}{(x+y)^{2}}}dxdy\,$

solution $\int _{1}^{4}\int _{{0}}^{{{\sqrt {4-x}}}}xydxdy\,$

solution $\int _{1}^{2}\int _{{x}}^{{x{\sqrt {3}}}}xydxdy\,$

solution $\int _{1}^{2}\int _{1}^{x}xy^{2}dxdy\,$

solution $\int _{{0}}^{{{\frac {\pi }{4}}}}\int _{{0}}^{{{\frac {\pi }{2}}}}\sin(x+y)dxdy\,$

solution $\int _{0}^{a}\int _{{0}}^{{{\sqrt {a^{2}-x^{2}}}}}y^{3}dydx\,$

solution $\int _{0}^{1}\int _{{{\sqrt {y}}}}^{{2-y}}x^{2}dxdy\,$

solution $\int _{0}^{2}\int _{{x^{2}}}^{{2x}}(2x+3y)dydx\,$

solution $\int _{0}^{a}\int _{{0}}^{{{\sqrt {a^{2}-x^{2}}}}}xydxdy\,$

solution $\int _{0}^{1}\int _{{0}}^{{1-x}}(x^{2}+y^{2})dydx\,$

solution $\int _{0}^{a}\int _{{{\frac {x^{2}}{a}}}}^{{2a-x}}xydydx\,$

solution $\int _{0}^{1}\int _{{-{\sqrt {y}}}}^{{{\sqrt {y}}}}dxdy+\int _{1}^{9}\int _{{{\frac {y-3}{2}}}}^{{{\sqrt {y}}}}dxdy\,$

solution $\int _{{0}}^{{2a}}\int _{{{\frac {y^{2}}{4a}}}}^{{3a-y}}(x^{2}+y^{2})dxdy\,$

solution Evaluate $\iint _{R}xydxdy\,$ where R is the positive quadrant of the circle $x^{2}+y^{2}=a^{2}\,$

solution Evaluate the double integral $\iint xy(x+y)dxdy\,$ over the region bounded by the curves $y=x,y=x^{2}\,$

solution Evaluate $\int _{{0}}^{{2a}}\int _{{0}}^{{{\sqrt {2ax-x^{2}}}}}(x^{2}+y^{2})dxdy\,$ by changing into polar coordinates.

solution Evaluate $\iint xy(x^{2}+y^{2})^{{{\frac {n}{2}}}}dxdy\,$ over the positive quadrant of the circle $x^{2}+y^{2}=a^{2}\,$ supposing n+3>0.

solution Find $\iint _{R}xydxdy\,$ where R is the region bounded by $x=1,x=2,y=0,xy=1\,$

solution Evaluate $\int _{{1}}^{{\log 8}}\int _{{0}}^{{\log y}}e^{{x+y}}dxdy\,$

solution Evaluate $I=\iint _{D}(x^{2}+y^{2})dxdy\,$ where D is bounded by $y=x\,$ and $y^{2}=4x\,$

solution $\iint _{D}(4xy-y^{2})dxdy\,$ where D is the reactangle bounded by $x=1,x=2,y=0,y=3\,$

solution $\iint _{D}(x^{2}+y^{2})dxdy\,$ where D is the region bounded by $y=x,y=2x,x=1\,$ in the first quadrant.

solution $\iint _{D}(1+x+y)dxdy\,$ where D is the region bounded by the lines $y=-x,x={\sqrt {y}},y=2,y=0\,$

solution $\iint _{D}xydxdy\,$ where D is the domain bounded by the parabola $x^{2}=4ay\,$,the ordinates $x=a\,$ and x-axis.

solution Evaluate $\iint (x-y)dxdy\,$ over the region between the line $x=y\,$ and the parabola $y=x^{2}\,$

solution Find the area bounded by the ellipse ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1\,$

solution $I_{D}=\iint _{D}x^{3}ydxdy\,$ where D is the region enclosed by the ellipse ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1\,$ in the first quadrant.

solution Find by double integration,the area which lies inside the cardoid $r=a(1+\cos \theta )\,$ and outside the circle r=a.

solution Find the area in the XY-plane bounded by the lemniscate $r^{2}=a^{2}\cos 2\theta \,$

solution Find the area bounded by the curves $y^{2}=x^{3}\,$ and $x^{2}=y^{3}\,$

solution Find the area of the domains $3x=4-y^{2},x=y^{2}\,$ in the XY-plane.

solution Find the area of the region bounded by the parabola $y^{2}=4ax\,$ and the straight line $x+y=3a\,$ in the XY-plane.

solution Find the area common to the parabolas $y^{2}=4a(x+a),y^{2}=4b(b-x)\,$

solution Find the area of the domains $x=y-y^{2},x+y=0\,$ in the XY-plane.

solution Find the area of the domains $3y^{2}=25x,5x^{2}=9y\,$ in the XY-plane.

solution Find the mass,coordinates of the centre of gravity and moments of inertia relative to x-axis,y-axis and origin of a reactangle $0\leq x\leq 4,0\leq y\leq 2\,$ having mass density xy.

solution Find the volume of tetrahedron in space cut from the first octant by the plane $6x+3y+2z=6\,$

solution Calculate the volume of a solid whose base is in a xy-plane and is bounded by the parabola $y=4-x^{2}\,$ and the straight line $y=3x\,$,while the top of the solid is in the plane $z=x+4\,$.

solution Find the moment of inertia of the area bounded by the circle $x^{2}+y^{2}=a^{2}\,$ in the first quadrant,assume the surface density of 1.

solution A plane lamina of non uniform density is in the form of a quadrant of the ellipse ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1\,$.If the density at any point (x,y)be kxy,where K is a constant,find the coordinates of the centroid of the lamina.

solution Determine the volume of the space below the paraboloid $x^{2}+y^{2}+z-4=0\,$ and above the square in the xy-plane with vertices at $(0,0),(0,1),(1,0),(1,1)\,$.

solution Find the volume of the solid under the surface $az=x^{2}+y^{2}\,$ and whose base R is the circle $x^{2}+y^{2}=a^{2}\,$.

solution Find the volume enclosed by the coordinate planes and that portion of the plane $x+y+z=1\,$ which lies in the first quadrant.

solution A circular hole of radius b is made centrally through a sphere of radius a.Find the volume of the remaining portion of the sphere.

solution Find the volume of the region bounded by the paraboloids $z=x^{2}+y^{2}\,$ and $z={\frac {6-{\frac {x^{2}+y^{2}}{2}}}\,}$.

solution Find the volume of the region bounded by the paraboloid $z=x^{2}+y^{2}\,$ and the plane z=4.

solution Find the volume of the solid enclosed by the ellipsoid ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1\,$

solutionFind the volume of the region in space bounded by the surface $z=1-(x^{2}+y^{2})\,$ on the sides by the planes $x=0,y=0,x+y=1\,$ and below by the plane z=0.

solution Evaluate $\int _{0}^{1}\int _{0}^{x}\int _{{0}}^{{x+y}}(x+y+z)dzdydx\,$

solution Find the volume bounded by the ellipsoidic paraboloids $z=x^{2}+9y^{2}\,$ and $z=18-x^{2}-9y^{2}\,$

solution Find the total mass of the region in the cube $0\leq x\leq 1,0\leq y\leq 1,0\leq z\leq 1\,$ with density at any point given by $xyz\,$

solution Find the mass,centroid of the tetrahedron bounded by the coordinate planes and the plane ${\frac {x}{a}}+{\frac {y}{b}}+{\frac {z}{c}}=1\,$

solution Evaluate $\int _{0}^{2}\int _{1}^{z}\int _{{0}}^{{yz}}xyzdxdydz\,$

solution If the radius of the base and altitude of a right circular cone are given by a and h respectively,express its volume as a tripple integral and evaluate it using cylindrical coordinates.

solution Evaluate $\int _{0}^{a}\int _{0}^{x}\int _{{0}}^{{y+x}}e^{{x+y+z}}dzdydx\,$

solution Find the volume bounded by the sphere $x^{2}+y^{2}+z^{2}=a^{2}\,$
solution Evaluate $\iiint xyzdxdydz\,$ over the positive octant of the sphere $x^{2}+y^{2}+z^{2}=a^{2}\,$

solution Assuming $\rho (x,y,z)=1\,$,find the centroid of the portion of the sphere $x^{2}+y^{2}+z^{2}=a^{2}\,$ in the first octant.

solution Find the mass and moment of inertia of a sphere of radius 'a' with respect to a diameter if the density is proportional to the distance from the center.

solution Evaluate $\iiint _{E}y^{2}x^{2}dV\,$ Where E is the region bounded by the paraboloid $x=1-y^{2}-z^{2}\,$ and the plane $\,x=0$

solution $\int _{{-2}}^{{2}}\int _{{0}}^{{{\sqrt {4-y^{2}}}}}\int _{{-{\sqrt {4-x^{2}-y^{2}}}}}^{{{\sqrt {4-x^{2}-y^{2}}}}}y^{2}{\sqrt {x^{2}+y^{2}+z^{2}}}dzdxdy\,$ Evaluate using spherical coordinates

solution Evaluate $\iint _{{R}}(x+y)e^{{x^{2}-y^{2}}}dA\,$ Where $_{{R}}\,$ is the rectangle enclosed by the lines $x-y=0,x-y=2,x+y=0,x+y=3\,$