Question

the function ( f ), shown below, is comprised of a semi - circle and piecewise linear segments. what is the value of ( int_{-1}^{9} f(x) dx )? write your answer in simplest form.

Explanation:

Step1: Split integral into 3 intervals

$\int_{-1}^{9} f(x) dx = \int_{-1}^{0} f(x) dx + \int_{0}^{6} f(x) dx + \int_{6}^{9} f(x) dx$

Step2: Calculate $\int_{-1}^{0} f(x) dx$

This is a triangle below x-axis, base $1$, height $-2$.
Area = $\frac{1}{2} \times 1 \times (-2) = -1$

Step3: Calculate $\int_{0}^{6} f(x) dx$

This is a semicircle, radius $3$.
Area = $\frac{1}{2} \pi r^2 = \frac{1}{2} \pi (3)^2 = \frac{9\pi}{2}$

Step4: Calculate $\int_{6}^{9} f(x) dx$

This is a triangle below x-axis, base $3$, height $-3$.
Area = $\frac{1}{2} \times 3 \times (-3) = -\frac{9}{2}$

Step5: Sum all 3 results

$-1 + \frac{9\pi}{2} - \frac{9}{2} = \frac{9\pi}{2} - \frac{11}{2}$

Answer:

$\frac{9\pi - 11}{2}$