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Calc 3 13.4

 

<== Calculus 3
Number Question Answer
See the work
1. Let C be the positively oriented circle x^{2} + y^{2} = 1. Use Green's Theorem to evaluate the line integral \int_{C} 15 y \, dx + 17 x \, dy.

6.2832 The work
2. Let C be the positively oriented square with vertices ( 0, 0 ), ( 3, 0 ), ( 3, 3 ), ( 0, 3 ). Use Green's Theorem to evaluate the line integral \int_{C} 1 y^{2} x \, dx + 7 x^{2}
y \, dy.

243 The work
3. A)
Use Green's theorem to compute the area inside the ellipse \displaystyle rac{x^2}{7^2} + rac{y^2}{12^2} = 1.
Use the fact that the area can be written as

\iint_D dx\,dy  = rac12\int_{\partial D}-y\ dx + x\ dy \ .
Hint: x(t) = 7 \cos(t).
The area is .

B)
Find a parametrization of the curve x^{2/3} + y^{2/3} = 7^{2/3} and use it to compute the area of the interior.
Hint: x(t) = 7 \cos^3(t).

A) 263.89

B) 57.73
The work
4. Use Green's theorem to compute the area inside the ellipse \displaystyle rac{x^2}{5^2} + rac{y^2}{3^2} = 1.
That is use the fact that the area can be written as

\iint_D dx\,dy = \iint_D (rac{\partial Q}{\partial x} - rac{\partial P}{\partial y}) dx \,dy = \int_{\partial D}P dx + Qdy
for appropriately chosen P and Q.
The area is .
47.124 The work
5. Use Green's theorem to compute the area of one petal of the 36-leafed rose defined by r = 7sin(18&theta).
It may be useful for recall that the area of a region D enclosed by a curve C can be expressed as A = rac{1}{2}\int_C x\,dy - y\,dx.
2.138 The work
6. question Answer The work
7. question Answer The work