' 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 =

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

calculate the double integral

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 =

How would you evaluate the double integral of a continuous function ƒ(x, y) over the region R in the xy-plane enclosed by the tri-angle with vertices (0, 1), (2, 0), and (1, 2)? Give reasons for your answer.

21 please help

Suppose that f(x, y) = D Y 1 + x at which {(x, y) | 0 ≤ x ≤ 2, - x ≤ y ≤ √T}. Then the double integral of f(x, y) over Dis [ f(x, y)dxdy =