b52bae…: “Through the red rectangle”

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Suppose $f:\R \to \R$ is a function with the property that: $\int _0^4 f(t) \d t = \pi$ Let $C = \{(x,y): x^2+y^2 \le 4\}$ . Compute: $\iint _C f(x^2 + y^2) \d A \begin {prompt} = \answer {\pi ^2} \end {prompt}$

Start by converting to polar coordinates.

Write with me: $\iint _C f(x^2 + y^2) \d A = \int _0^{\answer {2\pi }}\int _0^{\answer {2}} f(r^2) r \d r \d \theta$

Now let $t = r^2$

So: $\begin{align*} t &= \answer{r^2}\\ \d t &= \answer{2 r} \d r\\ \d r &= \frac{\d t}{\answer{2r}} \end{align*}$

Now we have: $\int _0^{\answer {2\pi }}\int _0^{\answer {2}} f(r^2) r \d r \d \theta = \int _0^{2\pi }\int _0^{\answer {4}} \frac {f(t)}{\answer {2}} \d t \d \theta$