VC4.8

From Example Problems
Jump to: navigation, search

Since r=xi+yj,dr=dxi+dyj\,

\int _{{C}}F\cdot \,dr=\int _{{C}}[xyi+(x^{2}+y^{2})j]\cdot (dxi+dyj)\,

=\int _{{C}}(xydx+(x^{2}+y^{2})dy)=\int _{{C}}xydx+\int _{{C}}(x^{2}+y^{2})dy\,

=\int _{{2}}^{{4}}x(x^{2}-4)dx+\int _{{0}}^{{12}}[(y+4)+y^{2}]dy\, [Since y=x^{2}-4\,,also for the given curve C,it is given that x varies from 2 to 4 and y varies from 0 to 12]

=[{\frac  {x^{4}}{4}}-2x^{2}]_{{2}}^{{4}}+[{\frac  {y^{2}}{2}}+4y+{\frac  {y^{3}}{3}}]_{{0}}^{{12}}=[(64-32)-(4-81)]+[(72+48+576)-0]=732\,

Main Page