Why is a change in limits of integration necessary here?

Question

Why did they change the limits and multiply by two? From what I can see the function isn't symmetrical to the x axis for the last few steps (https://i.stack.imgur.com/QtX5z.png)

Answer 1

If $f(x)=f(-x)$, the with $u=-x$, $du=-dx$, you have $$ \int_{-a}^0f(x)\,dx=-\int_a^0 f(u)\,du=\int_0^a f(u)\,du. $$ So $$ \int_{-a}^af(x)\,dx=\int_{-a}^0f(x)\,dx+\int_0^af(x)\,dx =\int_0^af(x)\,dx+\int_0^af(x)\,dx=2\int_0^af(x)\,dx. $$

What happens in your example is that first one does the above, and only then you can write $(\sin^2\theta)^{3/2}=\sin^3\theta$. Remember that $\sqrt{x^2}=|x|$. Here, after the symmetry trick allows you to have $\theta\geq0$, one can remove the absolute value (or, equivalently, say that $(\sin^2\theta)^{3/2}=\sin^3\theta$).