Double integrals of exponential functions

Question

I need to find the double integral of $$e^{\frac{x}{y^2}}$$ bound by the $y\mbox{-axis}$, $x=y^2$, $y=1$, and $y=2$. The limits of integration were easy to find, but I am pretty confused about how to to treat exponential functions with multiple variables in the exponent. Any help would be greatly appreciated!

Answer 1

Treat $y^2$ as a constant; don't worry if it has an exponent or anything like that.

$$ \begin{aligned} \ \int_1^2 \int_0^{y^2} \exp\left(x/y^2\right) dx\, dy &= \int_1^2 \left[ y^2\exp \left(x/y^2\right) \right]_0^{y^2} dy \\ \end{aligned} $$

What do you get from there?

Note: In general, exponential functions with an exponent in their argument are pretty messy, at least if there's nothing else to integrate. What we hope happens with this kind of problem is that something either cancels or, probably more likely, something appears — for example, after integrating with respect to one variable, perhaps you end up with an integral like

$$\int x^2 \exp\left(x^3\right) dx$$