Question

Evaluate. You might consider changing the order of integration.

Answer 1

\[\int\limits_{0}^{4}\int\limits_{0}^{1}\int\limits_{2y}^{2}\frac{ 4\cos(x ^{2)} }{ 2\sqrt{z} } dxdydz\]

Answer 2

As a start : sketch the region

Answer 3

you need to change order of integration as the integral cos(x^2) is not elementary, you don't have a choice

Answer 4

\[2y\le x\le 2\]
\[0\le y\le 1\]

so \[0\le 2y\le x\le 2\]

and

\[0\le y\le x/2\le 1\]

Answer 5

wow! looks nice

Answer 6

region in xy plane is sufficient however

Answer 7

Thanks lol, but that is right?

Answer 8

thats perfect ! thats the space overwhich we're integrating the given function

Answer 9

lets see the picture in xy plane

Answer 10

How do we know which limits of integration to switch?

Answer 11

we know for sure that we cannot evaluate cos(x^2) dx

Answer 12

so lets switch dxdy to dydx and see what we get