Question: The graph of \( f \) is shown. Evaluate each integral by interpreting it in terms of areas. \[…

The graph of \( f \) is shown. Evaluate each integral by interpreting it in terms of areas.

\[ \begin{array}{l} \text{(a)} \quad \int_{0}^{18} f(x) \, dx \\ \text{(b)} \quad \int_{0}^{45} f(x) \, dx \\ \text{(c)} \quad \int_{45}^{63} f(x) \, dx \\ \text{(d)} \quad \int_{1}^{51} f(x) \, dx \end{array} \]

The graph shows a function \( y = f(x) \) with the following points:

• \( (0, 0) \)
• \( (18, 18) \)
• \( (36, 36) \)
• \( (54, 0) \)
• \( (72, 18) \)

The area under the curve is divided into different sections, and the integrals are evaluated based on these areas.

Solution

Evaluating the Integrals Based on the Graph of \( f(x) \): (b) \(\int_0^{45} f(x) \, dx\) To evaluate the integral from \( x = 0 \) to \( x = 45 \), we can break it down into three segments based on the behavior of \( f(x) \): 1. From \( x = 0 \) to \( x = 18 \): \[ \int_0^{18} f(x) \, dx = 324 \] 2. From \( x = 18 \) to \( x = 36 \): Since \( f(x) = 18 \) in this interval, \[ \int_{18}^{36} f(x) \, dx = 18 \times (36 - 18) = 324 \] 3. From \( x = 36 \) to \( x = 45 \): In this interval, \( f(x) \) decreases linearly from \( 18 \) at \( x = 36 \) to \( 12 \) at \( x = 45 \). The area can be calculated as the area of a trapezoid: \[ \int_{36}^{45} f(x) \, dx = \frac{(18 + 12)}{2} \times (45 - 36) = \frac{30}{2} \times 9 = 15 \times 9 = 135 \] Total Integral from \( 0 \) to \( 45 \): \[ \int_0^{45} f(x) \, dx = 324 + 324 + 135 = 783 \] --- (c) \(\int_{45}^{63} f(x) \, dx\) In this interval, \( f(x) \) decreases linearly from \( 12 \) at \( x = 45 \) to \( 0 \) at \( x = 63 \). The area can be calculated as the area of a trapezoid: \[ \int_{45}^{63} f(x) \, dx = \frac{(12 + 0)}{2} \times (63 - 45) = \frac{12}{2} \times 18 = 6 \times 18 = 108 \] --- (d) \(\int_{51}^{51} f(x) \, dx\) Since the lower and upper limits of the integral are the same, the integral evaluates to zero: \[ \int_{51}^{51} f(x) \, dx = 0 \]