MvCalc30

From Example Problems
Jump to: navigation, search

we find out the limits of x and y from the regions given x=y,y=x^{2}\,

x^{2}-x=0,x(x-1)=0,x=0,x=1\,

Hence the limits of x are from 0 to 1 and y are from x to x^{2}\,

The given integral is \iint _{R}(x-y)dxdy=\int _{0}^{1}[\int _{{x}}^{{x^{2}}}(x-y)dy]dx\,

=\int _{0}^{1}[xy-{\frac  {y^{2}}{2}}]_{{x}}^{{x^{2}}}dx\,

=\int _{0}^{1}[(x^{3}-{\frac  {x^{4}}{2}})-(x^{2}-{\frac  {x^{2}}{2}})]dx\,

=\int _{0}^{1}[x^{3}-{\frac  {x^{4}}{2}}-{\frac  {x^{2}}{2}}]dx\,

=[{\frac  {x^{4}}{4}}-{\frac  {x^{5}}{10}}-{\frac  {x^{3}}{6}}]_{0}^{1}\,

={\frac  {1}{4}}-{\frac  {1}{10}}-{\frac  {1}{6}}={\frac  {15-6-10}{60}}={\frac  {-1}{60}}={\frac  {1}{60}}\, [Since the region of parabola is positive]

Main Page