Submit neatly written solutions to the following questions. You do not have to t
ID: 2884349 • Letter: S
Question
Submit neatly written solutions to the following questions. You do not have to type your solutions, just submit solutions that are clear, and easily read. Show sufficient work for full credit. Use Green's Theorem to set up and evaluate the vector line integral of F vector = (xe^-2x, x^4 + 2x^y^2), over the closed curve C, where C is the bound of the region between x^2 + y^2 = 1 and x^2 + y^2 = 4 for which x greaterthanorequalto 0. (In other words evaluate integral_C xe^-2x dx + (x^4 + 2x^2y^2) dy by using Green's Theorem.)Explanation / Answer
Solution:
Letting R be the region between these circles, we have
c [xe-2x dx + (x4 + 2x2 y2) dy]
= R [(/x) (x4 + 2x2 y2) - (/y)(xe-2x] dA, by Green's Theorem
= R (4x3 + 4xy2) dA
= R 4x(x2 + y2) dA
= ( = 0 to 2) (r = 1 to 2) 4(r cos ) * r^2 * (r dr d), via polar coordinates
= ( = 0 to 2) cos d * (r = 1 to 2) 4r^4 dr
= -sin() {from ( = 0 to 2)} * (r = 1 to 2) 4r^4 dr
= 0, due to the first factor.