Question

Line Integrals & Green's Theorem ("Take-Home" Problem) 13. (a) (4 pts) Set up and evaluate the line integral ?_C F · dr, where C is the circle x^2 + y^2 = 9 (oriented counterclockwise) and F(x,y) = y î + 2x ?. Evaluate this line integral directly — that is, find a vector function r(t) to parameterize the circle, compute dr and F(r(t)), and find limits of integration — do not use Green's Theorem. You may use a calculator or app to evaluate your integral once it is set up. (b) (4 pts) Suppose C is any closed curve C in the xy-plane (oriented counterclockwise), and D is the region in the xy-plane enclosed by C. For which of the following vector fields F(x,y) will the line integral ?_C F · dr equal the area of the region D? (Hint: Use Green's theorem.) Circle all correct answers. i. F(x,y) = y î + 2x ? ii. F(x,y) = -y/2 î + x/2 ? iii. F(x,y) = x î + y ? iv. F(x,y) = 0 î + x ? v. None of the above. (c) (2 pts) Is it possible to find a conservative vector field so that ?_C F · dr = the area of the region D enclosed by the closed curve C? Either give an example of a conservative field that works, or explain why it is not possible to find such a field.

          Line Integrals & Green's Theorem ("Take-Home" Problem)
13. (a) (4 pts) Set up and evaluate the line integral ?_C F · dr, where C is the circle x^2 + y^2 = 9 (oriented counterclockwise) and F(x,y) = y î + 2x ?.

Evaluate this line integral directly — that is, find a vector function r(t) to parameterize the circle, compute dr and F(r(t)), and find limits of integration — do not use Green's Theorem. You may use a calculator or app to evaluate your integral once it is set up.

(b) (4 pts) Suppose C is any closed curve C in the xy-plane (oriented counterclockwise), and D is the region in the xy-plane enclosed by C. For which of the following vector fields F(x,y) will the line integral ?_C F · dr equal the area of the region D? (Hint: Use Green's theorem.) Circle all correct answers.
i. F(x,y) = y î + 2x ?
ii. F(x,y) = -y/2 î + x/2 ?
iii. F(x,y) = x î + y ?
iv. F(x,y) = 0 î + x ?
v. None of the above.

(c) (2 pts) Is it possible to find a conservative vector field so that ?_C F · dr = the area of the region D enclosed by the closed curve C? Either give an example of a conservative field that works, or explain why it is not possible to find such a field.

Line Integrals Green's Theorem ("Take-Home" Problem)
13. (a) (4 pts) Set up and evaluate the line integral ?C F · dr, where C is the circle x^2 + y^2 = 9 (oriented counterclockwise) and F(x,y) = y î + 2x ?.

Evaluate this line integral directly — that is, find a vector function r(t) to parameterize the circle, compute dr and F(r(t)), and find limits of integration — do not use Green's Theorem. You may use a calculator or app to evaluate your integral once it is set up.

(b) (4 pts) Suppose C is any closed curve C in the xy-plane (oriented counterclockwise), and D is the region in the xy-plane enclosed by C. For which of the following vector fields F(x,y) will the line integral ?C F · dr equal the area of the region D? (Hint: Use Green's theorem.) Circle all correct answers.
i. F(x,y) = y î + 2x ?
ii. F(x,y) = -y/2 î + x/2 ?
iii. F(x,y) = x î + y ?
iv. F(x,y) = 0 î + x ?
v. None of the above.

(c) (2 pts) Is it possible to find a conservative vector field so that ?C F · dr = the area of the region D enclosed by the closed curve C? Either give an example of a conservative field that works, or explain why it is not possible to find such a field.

Added by Marcos Z.

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