Question

Use green's theorem to find integral subscript c left parenthesis y space plus space e to the power of square root of x end exponent right parenthesis d x space plus space left parenthesis 2 x plus cos open parentheses y squared close parentheses right parenthesis d y, where c is the boundary of the region bounded by the parabola y = x2 and the lines y = 0 and x = 1. A right parenthesis space e plus cos left parenthesis 1 right parenthesis b right parenthesis space e minus cos left parenthesis 2 right parenthesis c right parenthesis space e to the power of 1 divided by 2 end exponent minus cos left parenthesis 4 right parenthesis d right parenthesis space e to the power of 1 divided by 2 end exponent plus cos left parenthesis 1 right parenthesis e right parenthesis space 1 third.

Answer 1

My best interpretation of the math here is that you're talking about the line integral,

$\displaystyle \int_C \left(y+e^{\sqrt x}\right) \, dx + \left(2x + \cos\left(y^2\right)\right) \, dy$

I won't bother trying to decipher what look like multiple choice solutions.

By Green's theorem, the line integral above is equivalent to

$\displaystyle \iint_D \frac{\partial\left(2x+\cos\left(y^2\right)\right)}{\partial x} - \frac{\partial\left(y+e^{\sqrt x}\right)}{\partial y} \, dx \, dy$

where D is the set

$D = \left\{ (x, y) : 0 \le x \le 1 \text{ and } 0 \le y \le x^2 \right\}$

Compute the double integral:

$\displaystyle \int_0^1 \int_0^{x^2} \left(2 - 1\right) \, dy \, dx = \int_0^1 \int_0^{x^2} dy \, dx = \int_0^1 x^2 \, dx = \boxed{\frac13}$