# 4.4.E: Green's Theorem (Exercises)

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Exercise $$\PageIndex{1}$$

Let $$D$$ be the closed rectangle in $$\mathbb{R}^2$$ with vertices at (0,0), (2,0), (2,4), and (0,4), with boundary $$\partial D$$ oriented counterclockwise. Use Green’s theorem to evaluate the following line integrals.

(a) $$\int_{\partial D} 2 x y d x+3 x^{2} d y$$

(b) $$\int_{\partial D} y d x+x d y$$

(a) $$\int_{\partial D} 2 x y d x+3 x^{2} d y=80$$

Exercise $$\PageIndex{2}$$

Let $$D$$ be the triangle in $$\mathbb{R}^2$$ with vertices at (0,0), (2,0), and (0,4), with boundary $$\partial D$$ oriented counterclockwise. Use Green’s theorem to evaluate the following line integrals.

(a) $$\int_{\partial D} 2 x y^{2} d x+4 x d y$$

(b) $$\int_{\partial D} y d x+x d y$$

(c) $$\int_{\partial D} y d x-x d y$$

(a) $$\int_{\partial D} 2 x y^{2} d x+4 x d y=\frac{16}{3} \text { (c) } \int_{\partial D} y d x-x d y=-8$$

Exercise $$\PageIndex{3}$$

Use Green’s theorem to find the area of a circle of radius $$r$$.

Exercise $$\PageIndex{4}$$

Use Green’s theorem to find the area of the region $$D$$ enclosed by the hypocycloid

$x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}} , \nonumber$

where $$a > 0$$. Note that we may parametrize this curve using

$\varphi(t)=\left(a \cos ^{3}(t), a \sin ^{3}(t)\right) , \nonumber$

$$0 \leq t \leq 2 \pi$$.

$$\frac{3}{8} \pi a^{2}$$

Exercise $$\PageIndex{5}$$

Use Green’s theorem to find the area of the region enclosed by one “petal” of the curve parametrized by

$\varphi(t)=(\sin (2 t) \cos (t), \sin (2 t) \sin (t)) . \nonumber$

$$\frac{\pi}{8}$$

Exercise $$\PageIndex{6}$$

Find the area of the region enclosed by the cardioid parametrized by

$\varphi(t)=((2+\cos (t)) \cos (t),(2+\cos (t)) \sin (t)) , \nonumber$

$$0 \leq t \leq 2 \pi$$.

$$\frac{9 \pi}{2}$$

Exercise $$\PageIndex{7}$$

Verify (4.4.23), thus completing the proof of Green’s theorem.

Exercise $$\PageIndex{8}$$

Suppose the vector field $$F: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$$ with coordinate functions $$p=F_{1}(x, y)$$ and $$q=F_{2}(x, y)$$ is $$C^1$$ on an open set containing the Type III region $$D$$. Moreover, suppose $$F$$ is the gradient of a scalar function $$f: \mathbb{R}^{2} \rightarrow \mathbb{R}$$.

(a) Show that

$\frac{\partial q}{\partial x}-\frac{\partial p}{\partial y}=0 \nonumber$

for all points $$(x,y)$$ in $$D$$.

(b) Use Green’s theorem to show that

$\int_{\partial D} p d x+q d y=0 , \nonumber$

where $$\partial D$$ is the boundary of $$D$$ with counterclockwise orientation.

Exercise $$\PageIndex{9}$$

How many ways do you know to calculate the area of a circle?

Exercise $$\PageIndex{10}$$

Who was George Green?

Exercise $$\PageIndex{11}$$

Explain how Green’s theorem is a generalization of the Fundamental Theorem of Integral Calculus.

Exercise $$\PageIndex{12}$$

Let $$b > a$$, let $$C_1$$ be the circle of radius $$b$$ centered at the origin, and let $$C_2$$ be the circle of radius $$a$$ centered at the origin. If $$D$$ is the annular region between $$C_1$$ and $$C_2$$ and $$F$$ is a $$C^1$$ vector field with coordinate functions $$p=F_{1}(x, y)$$ and $$q=F_{2}(x, y)$$, show that

$\iint_{D}\left(\frac{\partial q}{\partial x}-\frac{\partial p}{\partial y}\right) d x d y=\int_{C_{1}} p d x+q d y+\int_{C_{2}} p d x+q d y , \nonumber$

where $$C_1$$ is oriented in the counterclockwise direction and $$C_2$$ is oriented in the clockwise direction. (Hint: Decompose $$D$$ into Type III regions $$D_1$$, $$D_2$$, $$D_3$$, and $$D_4$$, each with boundary oriented counterclockwise, as shown in Figure 4.4.5.)

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