' Suppose that f(x, y) = e³/ on the domain D = {(x, y) | 0 ≤ y ≤ 2,0 ≤ x ≤ y³}. D Then the double integral of f(x, y) over D is [ [ f(x, y)dxdy =

Compute the real part of the integral [ Answer: f(z)dz where f(z) = u(x, y) + iv(x, y) with u(x, y) = x + y²,v(x, y) = y + x² and contour C is the straight line z(t) = (5.5-8.2i)r with t E (0, 1).

Show that the line integral | 2xe-" dx + (2y – a²e=") dy is independent of path and evaluate the integral where C is any path from (1,0) to (2, 1).

Verify Green's Theorem by evaluating both integrals y² ax + x² dy = x dy /R. ax ду for the given path. C: rectangle with vertices (0, 0), (6, 0), (6, 7), and (0, 7) y2 dx + x² dy : dA = R. dy

Evaluate both sides of the equation to verify Green's Theorem for the  integral (image attached) Where C: line segment from (0,0) to (2,0)                                      from (2,0) to (1,4)                                   and from (1,4) to (0,0)

Calculate the line integral f(3xy³ − 4x + 4y + 8) dx + (−3xy+5) dy, where C' is the rectangle with vertices (2, −4), (2, -3), (−3,−3), and (−3,-4) oriented clockwise. Enter an exact answer. Provide your answer below: f(3x³ 4x + 4y + 8) dx + (−3xy + 5) dy =