Question

the function $f$, shown below, is comprised of a semi - circle and piecewise linear segments. what is the value of $int_{-9}^{2} f(x) dx$? write your answer in simplest form.

Explanation:

Step1: Split integral into 3 parts

$\int_{-9}^{2} f(x) dx = \int_{-9}^{-3} f(x) dx + \int_{-3}^{0} f(x) dx + \int_{0}^{2} f(x) dx$

Step2: Calculate first integral (rectangle)

The function is constant $f(x)=-3$ on $[-9,-3]$.
Area = length $\times$ height: $\int_{-9}^{-3} -3 dx = (-3) \times (-3 - (-9)) = -3 \times 6 = -18$

Step3: Calculate second integral (semicircle)

This is a lower semicircle with radius 3. Area of full circle is $\pi r^2$, so semicircle area is $\frac{1}{2}\pi r^2$. Since it is below x-axis, the integral is negative:
$\int_{-3}^{0} f(x) dx = -\frac{1}{2}\pi (3)^2 = -\frac{9\pi}{2}$

Step4: Calculate third integral (rectangle)

The function is constant $f(x)=-3$ on $[0,2]$.
$\int_{0}^{2} -3 dx = (-3) \times (2 - 0) = -6$

Step5: Sum all 3 results

Total integral = $-18 - \frac{9\pi}{2} - 6$

Answer:

$-\frac{9\pi}{2} - 24$