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.

\[ \begin{array}{c|c} \text{y} & \text{x} \\ \hline 18 & 0 \\ 18 & 36 \\ 54 & 72 \\ \end{array} \]

(a) \(\int_{0}^{18} f(x) \, dx\)

324

(b) \(\int_{0}^{45} f(x) \, dx\)

(c) \(\int_{45}^{63} f(x) \, dx\)

(d) \(\int_{63}^{81} f(x) \, dx\)

Solution

To solve the integrals by interpreting them in terms of areas, we’ll consider the areas of geometric shapes formed under the graph of the function. (b) Evaluate \(\int_{0}^{45} f(x) \, dx\): 1. From \(0\) to \(18\), we have a triangle with base \(18\) and height \(18\). \[ \text{Area of triangle} = \frac{1}{2} \times 18 \times 18 = 162 \] 2. From \(18\) to \(36\), we have a rectangle with width \(18\) and height \(18\). \[ \text{Area of rectangle} = 18 \times 18 = 324 \] 3. From \(36\) to \(45\), we have a triangle with base \(9\) and height \(18\). \[ \text{Area of triangle} = \frac{1}{2} \times 9 \times 18 = 81 \] 4. Sum the areas: \[ \int_{0}^{45} f(x) \, dx = 162 + 324 + 81 = 567 \] (c) Evaluate \(\int_{45}^{63} f(x) \, dx\): 1. From \(45\) to \(54\), we have a triangle with base \(9\) and height \(18\). \[ \text{Area of triangle} = \frac{1}{2} \times 9 \times 18 = 81 \] Since it’s below the x-axis, it contributes negatively. \[ \text{Negative area} = -81 \] 2. From \(54\) to \(63\), we have a triangle with base \(9\) and height \(9\). \[ \text{Area of triangle} = \frac{1}{2} \times 9 \times 9 = 40.5 \] 3. Sum the areas: \[ \int_{45}^{63} f(x) \, dx = -81 + 40.5 = -40.5 \] The correct answers are: (b) \(\int_{0}^{45} f(x) \, dx = 567\) (c) \(\int_{45}^{63} f(x) \, dx = -40.5\)