Question

consider the graph of the function ( f(x) ):
evaluate the following integrals by interpreting them in terms of areas:
(a) ( int_{0}^{2} f(x) d x= )
(b) ( int_{0}^{5} f(x) d x= )
(c) ( int_{6}^{7} f(x) d x= )
(d) ( int_{0}^{9} f(x) d x= )
note: you can earn partial credit

Explanation:

Step1: Analyze the integral \(\int_{6}^{7}f(x)dx\)

The region from \(x = 6\) to \(x=7\) is a triangle. The base of the triangle \(b = 1\) and the height \(h=- 2\) (since it is below the \(x\) - axis). The area of a triangle is \(A=\frac{1}{2}bh\).

$$A=\frac{1}{2}(1)(-2)=- 1$$

Step2: Analyze the integral \(\int_{7}^{9}f(x)dx\)

The region from \(x = 7\) to \(x = 9\) is a trapezoid. The formula for the area of a trapezoid is \(A=\frac{1}{2}(a + b)h\), where \(a=-1\), \(b = 0\) and \(h = 2\)

$$A=\frac{1}{2}(-1+0)\times2=-1$$

Step3: Calculate \(\int_{6}^{9}f(x)dx=\int_{6}^{7}f(x)dx+\int_{7}^{9}f(x)dx\)

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Answer:

\(\int_{6}^{9}f(x)dx=-2\)