I'm trying to answer this question from my calculus 1 exam I did last December.
The question asks us to compute a definite integral by using the following change of variables $$ x=R \sin(w)$$ in the following equation: $$\int_{-R}^R \sqrt{R^2-x^2} \ dx $$ The thing that's confusing me is how to do this change of variable if the equation is already expressed in terms of $x$?
You define $x$ in terms of $w$ via the given rule. To make the change of variable you notice that $$\frac{dx}{dw}=R\cos(w)\iff dx=R\cos(w)\,dw$$ thus the integral becomes \begin{align*} \int_{-R}^R\sqrt{R^2-x^2}\,dx&=\int_{-\pi/2}^{\pi/2}\sqrt{R^2-R^2\sin^2(w)}\cdot R\cos(w)\,dw\\ &=\int_{-\pi/2}^{\pi/2}R\cos(w)\cdot R\cos(w)\,dw\\ &=R^2\int_{-\pi/2}^{\pi/2}\cos^2(w)\,dw \end{align*} you can end it from here.
