I'm working on the following problem:
Calculate $$ \iint_D x^2 y^4 dx dy$$ where $D$ is $\{ (x,y) \in \mathbb R^2 :x^4 < y < x^{1/8} \}$.
I'm familiar with how to solve double integrals, however I need help with how to come up with the upper and lower limits for the integrals. What confuses me is that there is only one interval with both x and y in it.
If you plot the graphs, $ y = x^4 $ and $ y = x^{1/8} $, it will be fairly clear what the region you are integrating over is, and it is also fairly simple to verify that any point in that region satisfies the inequality $ x^4 \lt y \lt x^{1/8} $.
To represent the region in the bounds of the integral, since we are integrating with dx first then dy, we would get $ y = x^{1/8} $ or $ x = y^8 $ as the upper bound, and similarly $ x = y^{1/4} $ as the lower bound. So we would get $$ \int_{0}^1\int_{y^{1/4}}^{y^8}x^2y^4dxdy $$
Hope this helps.