Question

evaluate the following integrals by interpreting them in terms of areas:
(a) $\int_{0}^{2} f(x) d x=4$
(b) $\int_{0}^{5} f(x) d x=10$
(c) $\int_{5}^{7} f(x) d x=-3$
(d) $\int_{0}^{9} f(x) d x=$

Explanation:

Step1: Calculate the area from \(x = 7\) to \(x = 9\)

The region from \(x = 7\) to \(x = 9\) is a triangle. The base \(b=2\) and the height \(h = 1\). The area of a triangle is \(A=\frac{1}{2}\times b\times h\). So \(A=\frac{1}{2}\times2\times1 = 1\)

Step2: Use the property of definite - integrals \(\int_{0}^{9}f(x)dx=\int_{0}^{5}f(x)dx+\int_{5}^{7}f(x)dx+\int_{7}^{9}f(x)dx\)

We know that \(\int_{0}^{5}f(x)dx = 10\), \(\int_{5}^{7}f(x)dx=- 3\) (negative because the region is below the \(x\) - axis), and \(\int_{7}^{9}f(x)dx = 1\) (positive because the region is above the \(x\) - axis)

Step3: Substitute the values into the formula

\(\int_{0}^{9}f(x)dx=10+( - 3)+1\)

Answer:

\(8\)