How do I sketch the region of integration and evaluate a double integral? 
How would I do this double integration and others like it? I don't necessarily want an exact answer to this one, but if you could go into detail on the steps I need to take, and how to take those steps I would really appreciate it!
For some background, I'm not very sure on my methods of integration, or when to use the few ones I do know.
 A: My personal recommendation for how to sketch double-and-so-on integrals' bounds:


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*First, we note what each integral is integrating with respect to. For this example, I'll be considering your left integral. The inner integral goes from $y = 0$ to $y = x^2$ (because of inner variable of integration being $dy$), and $x = 0$ to $x = 1$ (because the outer variable of integration is $dx$).

*Sketch those lines/equations in the plane.

*Sketch the intersection of these lines.
This is basically what I get for your example (my poor shading in black):

Okay, so where from here?
You basically integrate the inner integral ($\int_0^{x^2} x^2 y dy$) first - but you treat anything that isn't related to the variable of integration (your $d\text{(whatever)}$ as if it were a constant. For example, you can take that $x^2$ outside as if it were $\pi$ or $e$ or whatever. You can also plug in $x^2$ after you integrate the $y$ with no problem.
Then you take that and use it in the inner integral. This time, wherever $x$ is, you now treat it like a variable. Here it's just integrating as normal.
Basically multivariable integration becomes a matter of knowing what you're integrating over and what you treat as a constant and when. Sketching in and of itself doesn't seem particularly necessary for these integrals, but it'll become more useful when you want to switch the order of integration or whatever later on.
