[Math] Evaluate the integral by reversing the order of integration

integrationmultivariable-calculus

I'm having trouble solving for the new limits when I reverse the order of integration for the integral $$\int_{0}^1\int_{x}^1{e^{x\over y}}dydx$$
If someone could help me understand how to solve for the new limits, that would be great. I don't think I'll need help with the integration, it's just the setting up that gives me the most trouble.

Best Answer

Here is your region:

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Notice, it is bounded by the lines $y=0$ and $y=1$, so those are our bounds for $y$. Next, for any specific $y_0$, we are considering the $x$ values that range from $0$ to $y_0$, so our bounds for $x$ are $0<x<y$. Is that clear? The integral becomes:

$$\int_{0}^1\int_0^ye^{\frac{x}{y}}dxdy$$

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