Question

Evaluate ModifyingBelow Integral from nothing to nothing With Upper C xy dx plus left parenthesis x plus y right parenthesis dy ∫ C xy dx+(x+y)dy along the curve y equals 4 x squared y=4x2 from left parenthesis negative 3 comma 36 right parenthesis (−3,36) to left parenthesis 1 comma 4 right parenthesis (1,4). ModifyingBelow Integral from nothing to nothing With Upper C xy dx plus left parenthesis x plus y right parenthesis dy ∫ C xy dx+(x+y)dy equals = nothing ​(Type an integer or a simplified​ fraction.)

Answer 1

Parameterize the curve by

$\begin{cases}x(t)=t\\y(t)=4t^2\end{cases}$

with $-3\le t\le1$. Then

$\displaystyle\int_Cxy\,\mathrm dx+(x+y)\,\mathrm dy$

$=\displaystyle\int_{-3}^14t^3\,\mathrm dt+(t+4t^2)(8t\,\mathrm dt)$

$=\displaystyle\int_{-3}^1(8t^2+36t^3)\,\mathrm dt=\boxed{-\frac{1936}3}$