VC4.8

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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\,

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