Question

the function f, shown below, is comprised of a semi - circle and piecewise linear segments. what is the value of $\int_{-2}^{8}f(x)dx$? write your answer in simplest form.

Explanation:

Step1: Divide the integral

$$\int_{-2}^{8}f(x)dx=\int_{-2}^{3}f(x)dx+\int_{3}^{8}f(x)dx$$

Step2: Calculate the first integral (semicircle)

The semicircle has radius \(r = 3\) (from \(x=-2\) to \(x = 1\) is the diameter \(d = 3\)). The formula for the area of a semicircle is \(A=\frac{1}{2}\pi r^{2}\). Since the semicircle is below the \(x\) - axis, the value of \(\int_{-2}^{3}f(x)dx=-\frac{1}{2}\pi(3)^{2}=-\frac{9\pi}{2}\)

Step3: Calculate the second integral (triangle)

For \(\int_{3}^{8}f(x)dx\), we use the formula for the area of a triangle \(A=\frac{1}{2}bh\). The base \(b = 8 - 3=5\) and the height \(h\) (from \(y=-6\) to \(y = 0\)) is \(6\). So \(\int_{3}^{8}f(x)dx=\frac{1}{2}\times5\times6 = 15\)

Step4: Sum the two integrals

$$\int_{-2}^{8}f(x)dx=-\frac{9\pi}{2}+15$$

Answer:

\(15-\frac{9\pi}{2}\)