Question

Sketch the region of integration and evaluate the iterated integral by choosing a convenient order of integration: ∫(limits of integration: 2 on top, 0 on bottom)∫(limits of integration: 1 on top, x/2 on bottom) cos(y^2)dydx

Answer 1

See how there is a Y^2 INSIDE of the cosine? that makes it really really hard to evaluate.
If we could somehow make a Y appear on the OUTSIDE...

We would have a Y^2 inside, and a Y outside... allowing us to perform a nice easy U-substitution.

The way we'll do that issss.. we want to do the X integration first, so a y term will pop in to say hello.

Let's first draw the region, and see if we can effectively switch the order of integration.

Answer 2

\[\huge \int\limits_0^2 \; \int\limits_{x/2}^1 \cos(y^2)\;dy\;dx\]