Question

the graph of ( f ) is shown below. evaluate each integral by interpreting it in terms of areas.

1. ( int_{-1}^{2} f(x) d x=)
2. ( int_{2}^{4} f(x) d x=)
3. ( int_{-1}^{7} f(x) d x=)

Explanation:

Step1: Evaluate \(\int_{-1}^{2}f(x)dx\)

The region from \(x = - 1\) to \(x=2\) is a triangle. The base of the triangle \(b=3\) (since \(2-(-1)=3\)) and the height \(h = 2\) (the \(y\) - value at \(x=-1\)). The area of a triangle \(A=\frac{1}{2}\times base\times height\). So \(A=\frac{1}{2}\times3\times2 = 3\).

Step2: Evaluate \(\int_{2}^{4}f(x)dx\)

The region from \(x = 2\) to \(x = 4\) consists of a triangle below the \(x\) - axis. The base \(b = 2\) (\(4 - 2\)) and the height \(h=2\). The area of the triangle \(A=\frac{1}{2}\times2\times2=2\). Since it is below the \(x\) - axis, \(\int_{2}^{4}f(x)dx=- 2\)

Step3: Evaluate \(\int_{-1}^{7}f(x)dx\)

First, \(\int_{-1}^{2}f(x)dx = 3\) (from Step1). Then \(\int_{2}^{4}f(x)dx=-2\) (from Step2). The region from \(x = 4\) to \(x=7\):
The region from \(x = 4\) to \(x = 6\) is a semicircle with radius \(r = 1\). The area of a semicircle \(A_{semicircle}=\frac{1}{2}\pi r^{2}=\frac{1}{2}\pi(1)^{2}=\frac{\pi}{2}\). The region from \(x = 6\) to \(x = 7\) is a triangle with base \(b = 1\) and height \(h=3\). The area of the triangle \(A_{triangle}=\frac{1}{2}\times1\times3=\frac{3}{2}\)
\(\int_{-1}^{7}f(x)dx=\int_{-1}^{2}f(x)dx+\int_{2}^{4}f(x)dx+\int_{4}^{7}f(x)dx\)
\(=3-2+\frac{\pi}{2}+\frac{3}{2}\)
\(=1+\frac{\pi + 3}{2}=\frac{2+\pi + 3}{2}=\frac{\pi + 5}{2}\)

Answer:

1. \(3\)
2. \(-2\)
3. \(\frac{\pi + 5}{2}\)