Finding $\int_0^7 g(r)~dr$ from $ g(r)=\begin{cases} \sqrt{9-(r-3)^2}, & 0\le r<3 \\ r+2, & 3\le r<5 \\ \sqrt{4-(r-5)^2}, & 5 \le r \le 7\end{cases} $ I have some piece-wise integration that I for some reason am not getting.
$$
g(r) = 
\begin{cases}
\sqrt{9-(r-3)^2} && 0 \le r \lt 3 \\
r+2 && 3 \le r \lt 5 \\
\sqrt{4-(r-5)^2} && 5 \le r \le 7
\end{cases}
$$
I must find the area of $\int_0^7g(r) \ dr$.
I know that this must be equivalent to:
$$
\int_0^2\sqrt{9-(r-3)^2} \ dr \ + \ \int_3^4(r+2) \ dr \ + \ \int_5^7\sqrt{4-(r-5)^2} \ dr
$$
But my solution of $4.125 + 5.5 + \pi$ yields an incorrect answer?

I have no idea why you separated it like how you did it, this is incorrect:

$$\int_0^7 g(r)~dr= \int_0^2 \sqrt{9-(r-3)^2}~dr+\int_3^4 (r+2)~dr+\int_5^7 \sqrt{4-(r-5)^2}~dr$$

Doing your method will give this area instead:

Splitting definite integrals in two does not work like when you split summations!
$$\sum_{k=0}^n f(k)\equiv \sum_{k=0}^m f(k)+\sum_{k=m+1}^n f(k)$$
$$\int_0^n f(x)~dx\not\equiv \int_0^m f(x)~dx+\int_{m+1}^n f(x)~dx$$


In general, one can separate a definite integral into two definite ones as follows:
  $$\int_a^c f(x)~dx=\int_a^b f(x)~dx+\int_b^c f(x)~dx$$

Hence, you want to separate it as follows:
$$\begin{align} \int_0^7 g(r)~dr&=\int_0^3 \sqrt{9-(r-3)^2}~dr+\int_3^5 (r+2)~dr+\int_5^7 \sqrt{4-(r-5)^2}~dr \\&=\frac{9\pi}{4}+12+\pi\\&=\frac{13\pi}{4}+12\end{align}$$
Since you managed to evaluate your other integrals, I suppose you know how to evaluate these ones to $\frac{9\pi}{4}$ and $12$, so I left it as an exercise for the reader.

