Evaluate by interpreting as the area of a geometric figure (draw the figure and use geometric formula(s) to evaluate):\[\int_{0}^{5} (10 - 2x) \, dx = \]**Explanation:**This integral represents the area under the linear function \( f(x) = 10 - 2x \) from \( x = 0 \) to \( x = 5 \). 1. **Equation of the Line:** The equation \( f(x) = 10 - 2x \) is a straight line with a y-intercept at 10 and a slope of -2.2. **Geometric Interpretation:** - This line intersects the x-axis when \( 10 - 2x = 0 \), which gives \( x = 5 \). Therefore, the line forms a right triangle with the x-axis between \( x = 0 \) and \( x = 5 \). - The vertical height of the triangle at \( x = 0 \) is 10 (the y-intercept), and at \( x = 5 \), the height is 0 (where the line crosses the x-axis).3. **Calculating the Area:** - The base of the triangle is 5 (the distance between \( x = 0 \) and \( x = 5 \)). - The height of the triangle is 10. - The area \( A \) of a triangle is given by the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] - Substituting the values, we have: \[ A = \frac{1}{2} \times 5 \times 10 = 25 \]Thus, the value of the integral is 25. **Problem:**Evaluate by interpreting as the area of a geometric figure (draw the figure and use geometric formula(s) to evaluate):\[\int_{0}^{3} |x - 2| \, dx = \]**Solution Explanation:**To solve the integral \(\int_{0}^{3} |x - 2| \, dx\), we interpret it as calculating the area under the curve of the function \(y = |x - 2|\) from \(x = 0\) to \(x = 3\).### Steps:1. **Understanding the Function:** - The function \(y = |x - 2|\) is a V-shaped graph. It has a vertex at \(x = 2\). - For \(x < 2\), \(y = 2 - x\). - For \(x \geq 2\), \(y = x - 2\).2. **Evaluating the Integral:** - The interval \([0, 3]\) is split by \(x = 2\), where the nature of the absolute value function changes. - Compute the area from \(x = 0\) to \(x = 2\) and from \(x = 2\) to \(x = 3\).3. **Area Calculation:** - **From \(x = 0\) to \(x = 2\):** - The function is a line with negative slope, \(y = 2 - x\). - The area is a right triangle with base 2 and height 2. - Area = \(\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 2 = 2\). - **From \(x = 2\) to \(x = 3\):** - The function is a line with positive slope, \(y = x - 2\). - The area is a right triangle with base 1 (from 2 to 3) and height 1. - Area = \(\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 1 = 0.

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Answered: Evaluate by interpreting as the area of…

Evaluate by interpreting as the area of a geometric figure (draw the figure and use geometric formula(s) to evaluate): 5 [³ (10 (10 - 2x) dx =

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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