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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.)

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LammettHash

Parameterize the curve by

[tex]\begin{cases}x(t)=t\\y(t)=4t^2\end{cases}[/tex]

with [tex]-3\le t\le1[/tex]. Then

[tex]\displaystyle\int_Cxy\,\mathrm dx+(x+y)\,\mathrm dy[/tex]

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

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

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