Evaluate double integrals- check my work?

[mikky05v]

Homework Statement

Evaluate the iterated integrals (switch the order of integration if necessary)

I just need someone to check my work. My professor gave us this practice test to help study for our final but it isn't much use if I don't know if I'm doing it correctly.

I've been working them on my computer to make it easier to share them, If the images don't work please let me know.

The Attempt at a Solution

 

[Simon Bridge]

Notes: $\renewcommand{\dxy}{\;\text{d}x\text{d}y}$
Usually you can tell if you are doing them correctly by how they work out.
If you use LaTeX in posts, people can give better replies.

It also helps if you explain your reasoning - just naming the steps is good.
The idea is that we should be doing as little work as possible to see what you did.

For instance:
(f) $$\int_0^2 \int_{\frac{1}{2}x^2}^2 \sqrt{y}\cos y \dxy$$
step 1... change order of integration
... the trick here is to correctly identify the region of integration.
For us to check your work, we have to calculate it ourselves - you should think about telling us what you found.

Anyway - it looks OK from here.

(a) can be just integrated out - if you can differentiate an exponential you can do this.
you don't need to take the exponential outside the inner integration.

(b) can be integrated straight out as is - you have a redundant term.

(c) you had the right substitution - which was the trick here

(d) you combined integrals and changed the order of operations - don't know why you didn't just integrate them out.

(e) is the same as f but with different limits (and the roles of x and y are swapped over).
if you were brave you could have used the working for this as a template for f.

You seem to be finding complexity where there is none, so you are working harder than you need to - but it does mean you spot it when it's actually there. Glad to see you are not evaluating awkward numbers (like sin2 etc) but just leaving them - that's a good habit.