Rewriting sum of iterated integrals (order of integration)

[MuIotaTau]

Homework Statement

Rewrite the given sum of iterated integrals as a single iterated integral by reversing the order of integration, and evaluate.

$$\int_0^1 \int_0^x sin x dy dx + \int_1^2 \int_0^{2 - x} sin x dy dx$$

Homework Equations

None

The Attempt at a Solution

I drew the domains of each integral and saw that they appeared different, but I proceeded anyway. Changing the order of integration of each integral, I arrived at

$$\int_0^1 \int_y^1 sin x dx dy + \int_0^1 \int_1^{2-y} sin x dx dy$$

which doesn't appear to help me. This made me further suspicious about how I could possibly combine two integrals over two different domains. Any hints?

 

[UltrafastPED]

Assuming your result in (3) is correct:

your outer integral is the same for both terms ... so you can move it outside of the two inner integrals. But the inner integrals have the same integrands, and cover intervals (y,1) and (1,2-y) ... so they are contiguous.

That means they are equivalent to the same integrand over the union of the intervals: (y,2-y).

 

[MuIotaTau]

Assuming your result in (3) is correct:

your outer integral is the same for both terms ... so you can move it outside of the two inner integrals. But the inner integrals have the same integrands, and cover intervals (y,1) and (1,2-y) ... so they are contiguous.

That means they are equivalent to the same integrand over the union of the intervals: (y,2-y).

Oh wow, I never would have thought of that! That makes complete sense, though! Thank you so much, I really do appreciate it.