# A General Process for Finding Correct Bounds on Double Integral?

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?

• Instead going for $x$ first, he starts with going along the $y$-axis. Draw it, then you'll see what happens. – amsmath Oct 17 '17 at 4:12
• @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 – danielschnoll Oct 17 '17 at 4:19
• @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 – danielschnoll Oct 17 '17 at 4:19
• Forget about "general processes" in math. Usually, there is no such. – amsmath Oct 17 '17 at 4:21
• Have you drawn the picture? – amsmath Oct 17 '17 at 4:21

Here's two drawings that hopefully will be helpful for you.  