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Calculus 3 : Green's Theorem

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Example Questions

Example Question #1 : Green's Theorem

Use Green's Theorem to evaluate $\oint_C 5xy \ dx +x^3y^2 \ dy$ , where is a triangle with vertices (0,0) , (1,0) , (1,2) with positive orientation.

Possible Answers:

Correct answer:

Explanation:

First we need to make sure that the conditions for Green's Theorem are met.

The conditions are met because it is positively oriented, piecewise smooth, simple, and closed under the region (see below).

$0\leq x \leq 1$

$0 \leq y \leq 2x$

In this particular case P=5xy , and Q=x^3y^2 , where , and refer to $\oint_C P \ dx +Q \ dy$ .

We know from Green's Theorem that

$\oint_C P \ dx +Q \ dy=\int \int_D \Big(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\Big) dA$

So lets find the partial derivatives.

$\frac{\partial Q}{\partial x}=3x^2y^2$

$\frac{\partial P}{\partial y}=5x$

$\oint_C 5xy \ dx +x^3y^2 \ dy=\int \int_D 3x^2y^2-5x \ dA$

$=\int_{0}^{1} \int_{0}^{2x} 3x^2y^2-5x \ dy\ dx$

$=\int_{0}^{1} \Big(\frac{1}{3}\cdot3x^2y^3-5xy\Big)\Big|_{0}^{2x}\ dx$

$=\int_{0}^{1} \Big(x^2(2x)^3-5x(2x)\Big)\ dx$

$=\int_{0}^{1} 8x^5-10x^2\ dx$

$= \frac{4}{3}x^6-\frac{10}{3}x^3\Big|_{0}^{1}$

$= \frac{4}{3}(1)^6-\frac{10}{3}(1)^3-0=\frac{4}{3}-\frac{10}{3}=-\frac{6}{3}=-2$

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Example Question #2 : Green's Theorem

Use Green's Theorem to evaluate the line integral

$\int_{c}3ydx-4xdy$

over the region R, described by connecting the points (0,0),(2,0),(2,4) , orientated clockwise.

Possible Answers:

Correct answer:

Explanation:

Using Green's theorem

$\int_C{F}dr=\int \int\left[{\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}}\right]dA$

$\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}=-4-3=-7$

$Area=\frac{1}{2}bh=\frac{1}{2}\times2\times4=4$

since the region is oriented clockwise, we would have

$\int_C{F}dr=-\int\int\left[\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}\right]dA$

which gives us

$\int_C{F} dr=-(-7)\times4=28$

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Example Question #3 : Green's Theorem

Use Greens Theorem to evaluate the line integral

$\int_C 5y dx - 4x dy$

over the region connecting the points (0,0),(2,0),(2,4) oriented clockwise

Possible Answers:

Correct answer:

Explanation:

$\int_C 5y dx - 4x dy$

Using Green's theorem

$\int_C F dr=\int\int \left[\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}}\right]dA$

$Q=-4x\\ \frac{\partial{Q}}{\partial{x}}=-4$ $P=5y\\ \frac{\partial{P}}{\partial{y}}=5$

$Area=\frac{1}{2}bh=\frac{1}{2}\times2\times4=4$

Since the region is oriented clockwise

$\int_C F dr=-\int\int (-4-5)dA$

$\int_C F dr=\int\int 9dA=(9)(4)=36$

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Calculus 3 : Green's Theorem

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

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Example Question #1 : Green's Theorem

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Example Question #3 : Green's Theorem

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