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I'm asked to find the integral of the function $ xy e^{x^2 - y^2} $, on $$ D = {\{(x,y) \in R^2: 1 \leq y, \sqrt2 \le x \le 9, 1 \le x^2 - y^2 \le 9 \} } $$ I think I don't understand the necessary steps to get the bounds of integration. I know that $ x^2 - 9 \le y^2 \le x^2 - 1 $. I know that $y^2 \ge 1$ so we can get $ x = \sqrt 10 $. Since $ x \le 9 $, that means $ y^2 \le 80$, meaning $ y \le \sqrt80 $ Does that mean my final integral is the following? $$ \int_{\sqrt10}^9\int_{1}^{\sqrt80}f(x, y)dydx $$

FNB
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    Draw the domain $D$ and see what it looks like. Your bounds at the end are rectangular, but $D$ is not a rectangle. – Ninad Munshi Mar 09 '24 at 12:21
  • Oh, I feel dumb, I didn't think of drawing it... So it seems like my integral is 2 integrals: the first one is when $ \sqrt2 \le x \le \sqrt10 $, and $ 1 \le y \le \sqrt{x^2 - 1} $, and the second one is when $ \sqrt10 \le x \le 9 $ and $ \sqrt{x^2 - 9} \le y \le \sqrt{x^2 - 1} $? – FNB Mar 09 '24 at 12:32
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    That looks right to me – Ninad Munshi Mar 09 '24 at 12:34

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