Use Green's Theorem to evaluate the line integral along the positively oriented closed curve $C$.
$\oint _ { C } 3 x y d x + 4 x ^ { 2 } d y$, where $\mathrm { C }$ is the triangle with vertices $( 0,0 ) , ( 3,4 )$, and $( 0,4 )$.
A) 6
B) 60
C) 30
D) 66

Question

Quizplus · Accepted Answer

Green's Theorem states that

$$\oint _ { C } P d x + Q d y = \iint _ { D } \left( \frac { \partial Q } { \partial x } - \frac { \partial P } { \partial y } \right) d x d y$$,

where $D$ is the region enclosed by the closed curve $C$, and $(P,Q)$ is a vector field with continuous partial derivatives on $D$.

In this problem, we have $P = 3xy$ and $Q = 4x^2$. We can compute the partial derivatives as follows:

$$\begin{aligned} \frac{\partial Q}{\partial x} &= 8x \ \frac{\partial P}{\partial y} &= 3x \end{aligned}$$

Then, applying Green's Theorem, we have

$$\oint _ { C } 3 x y d x + 4 x ^ { 2 } d y = \iint _ { D } (8x-3x) d x d y = \iint _ { D } 5x dxdy$$

The region $D$ is the triangle with vertices $(0,0), (3,4), (0,4)$, and we can write it as the union of two smaller triangles: $\triangle_1 = \{(x,y) : 0 \leq x \leq 3, 0 \leq y \leq \frac{4}{3}x\}$ and $\triangle_2 = \{(x,y) : 0 \leq x \leq 3, \frac{4}{3}x \leq y \leq 4\}$.

Then, we have

$$\begin{aligned} \iint _ { D } 5x dxdy &= \iint _ { \triangle_1 } 5x dxdy + \iint _ { \triangle_2 } 5x dxdy \ &= \int _ { 0 } ^ { 3 } \int _ { 0 } ^ { \frac { 4 } { 3 } x } 5x dydx + \int _ { 0 } ^ { 3 } \int _ { \frac { 4 } { 3 } x } ^ { 4 } 5x dydx \ &= \int _ { 0 } ^ { 3 } \left[ 5xy \right] _ { y = 0 } ^ { y = \frac { 4 } { 3 } x } dx + \int _ { 0 } ^ { 3 } \left[ 5xy \right] _ { y = \frac { 4 } { 3 } x } ^ { y = 4 } dx \ &= \int _ { 0 } ^ { 3 } \frac { 20 } { 3 } x ^ { 2 } dx + \int _ { 0 } ^ { 3 } 20x - \frac { 80 } { 3 } x ^ { 2 } dx \ &= \frac { 20 } { 3 } \cdot \frac { 3 ^ { 3 } } { 3 } + \left[ 10x ^ { 2 } - \frac { 40 } { 3 } x ^ { 3 } \right] _ { 0 } ^ { 3 } \ &= 60 \end{aligned}$$

Therefore, the answer is C) 30.

Use Green's Theorem to Evaluate the Line Integral Along the Positively