You are given the four points in the plane \( A=(7,0) \), \( B=(12,1), C=(16,5) \), and \( D=(21,0) \). The graph of the function \( f(x) \) consists of the three line segments \( A B, B C \) and \( C D \). Find the integral \( \int_{7}^{21} f(x) d x \) by interpreting the integral in terms of sums of areas of elementary figures. \( \int_{7}^{21} f(x) d x= \)

To find the integral \( \int_{7}^{21} f(x) dx \) by interpreting it as the sum of areas of elementary figures, we first need to determine the shapes formed by the line segments connecting the given points \( A, B, C, \) and \( D \). 1. **Finding the equations of the line segments**: - **Segment \( AB \)**: Points \( A(7,0) \) and \( B(12,1) \). The slope is \( \frac{1-0}{12-7} = \frac{1}{5} \). The equation is \( f(x) = \frac{1}{5}(x - 7) \) for \( x \) in \([7, 12]\). - **Segment \( BC \)**: Points \( B(12,1) \) and \( C(16,5) \). The slope is \( \frac{5-1}{16-12} = 1 \). The equation is \( f(x) = x - 11 \) for \( x \) in \([12, 16]\). - **Segment \( CD \)**: Points \( C(16,5) \) and \( D(21,0) \). The slope is \( \frac{0-5}{21-16} = -1 \). The equation is \( f(x) = -x + 21 \) for \( x \) in \([16, 21]\). 2. **Calculating areas**: - The area under the curve from \( 7 \) to \( 12 \) (triangle \( \triangle AB \)): - Base = \( 5 \) (from \( 7 \) to \( 12 \)), Height = \( 1 \). \[ \text{Area}_{AB} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 5 \times 1 = 2.5. \] - The area from \( 12 \) to \( 16 \) (rectangle \( AB \)): - Width = \( 4 \) (from \( 12 \) to \( 16 \)), Height = \( 1 \). \[ \text{Area}_{BC} = \text{width} \times \text{height} = 4 \times 1 = 4. \] - The area from \( 16 \) to \( 21 \) (trapezoid \( \triangle CD \)): - Base1 = \( 5 \) (from point \( C \)), Base2 = \( 0 \) (from point \( D \)), Height = \( 5 \). - Average height = \( \frac{5 + 0}{2} = \frac{5}{2} \), Width = \( 5 \). \[ \text{Area}_{CD} = \text{Average base} \times \text{height} = 5 \times \frac{5}{2} = 12.5. \] 3. **Adding the areas**: \[ \int_{7}^{21} f(x) dx = \text{Area}_{AB} + \text{Area}_{BC} + \text{Area}_{CD} = 2.5 + 4 + 12.5 = 19. \] Thus, the value of the integral \( \int_{7}^{21} f(x) dx = 19 \).