Question

Reverse the order of integration and evaluate the integral (see question).

Answer 1

\[\int\limits_{0}^{2}\int\limits_{x}^{2}y^4 \cos(xy^2)dydx\]

Answer 2

It looks like we're integrating over a 2x2 square

Answer 3

a half of a 2x2 square...so we can just switch x and y in the bounds, because we have\[x\le y\le 2\]\[0\le x\le2\]

Answer 4

may I ask that by saying reversing the integration, you mean instead of dydx we'll do dxdy?

Answer 5

yeah, exactly

Answer 6

ah

Answer 7

so we see that
\[0 \le y \le 2\]
\[y \le x \le 2\]
so reversing the order of integration we'll have
\[\int\limits_{0}^{2}\int\limits_{y}^{2} y^4\cos(xy^2)dxdy\]