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So I'm working on some multivariable calculus homework, and I can't seem to figure out why my professor takes this particular approach to the solution...

The Question: $$ S = \{(x,y) \in R^2: 0 \leq x \leq 1, 0 \leq y \leq sin^{-1}x\} $$ And we have to evaluate $\int \int_{S} dA $

My professor's approach to this problem involves changing the integral bounds, so instead of the double integral setup looking like: $\int_{0}^{1} \int_{0}^{sin^{-1}}dydx$, it looks like $\int_{0}^{\pi/2} \int_{sin(y)}^{1}dxdy$

Can someone please explain how he got to this rearranged integral bounds setup, and additionally is there a general process for rewriting the integral bounds for a double integral?

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  • $\begingroup$ Instead going for $x$ first, he starts with going along the $y$-axis. Draw it, then you'll see what happens. $\endgroup$ – amsmath Oct 17 '17 at 4:12
  • $\begingroup$ @amsmath I realize he went for the y axis first, but I don't understand how he can go from a 1 to a $\pi/2$, or why the inverse sine becomes the lower bound of the integral $\endgroup$ – danielschnoll Oct 17 '17 at 4:19
  • $\begingroup$ @amsmath And also I was wondering if there is some sort of general process for approaching double integrals where you'd have to switch the bounds $\endgroup$ – danielschnoll Oct 17 '17 at 4:19
  • $\begingroup$ Forget about "general processes" in math. Usually, there is no such. $\endgroup$ – amsmath Oct 17 '17 at 4:21
  • $\begingroup$ Have you drawn the picture? $\endgroup$ – amsmath Oct 17 '17 at 4:21
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Here's two drawings that hopefully will be helpful for you.enter image description here

enter image description here

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