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Find the total area of the shaded region. Ay 24- 22- 20 18- 16- 144 12 10- 8- 6 y=x y=12 Ту *² 48 8 10 12 14 16 18 20 22 24 X

Find the total area of the shaded region. Ay 24- 22- 20 18- 16- 144 12 10- 8- 6 y=x y=12 Ту *² 48 8 10 12 14 16 18 20 22 24 X

Calculus: Early Transcendentals
Calculus: Early Transcendentals
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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**Problem Statement:** Find the total area of the shaded region. **Graph Explanation:** The graph is on a coordinate plane with the x-axis and y-axis labeled. The grid consists of squares for precise measurement. The shaded region is between three curves: 1. **Line \( y = x \):** - This is a straight line inclining upward from the origin at a 45-degree angle, represented in blue. 2. **Line \( y = 12 \):** - This is a horizontal line cutting across the graph at \( y = 12 \). 3. **Curve \( y = \frac{x^2}{48} \):** - This is a parabola with its vertex at the origin, opening upwards, represented in red. **Boundary Intersections:** - The line \( y = x \) and the parabola \( y = \frac{x^2}{48} \) intersect at \( x = 0 \). - The line \( y = 12 \) intersects \( y = x \) at \( (12, 12) \). - The parabola \( y = \frac{x^2}{48} \) intersects \( y = 12 \) at \( (24, 12) \). **Shaded Region:** The shaded area is bounded by these segments: - From \( x = 0 \) to \( x = 12 \), the area between the line \( y = x \) and \( y = 12 \). - From \( x = 12 \) to \( x = 24 \), the area between the parabola \( y = \frac{x^2}{48} \) and \( y = 12 \). To find the total area, calculate the area under \( y = 12 \) and subtract the areas under both \( y = x \) and \( y = \frac{x^2}{48} \) within their respective intervals.
Transcribed Image Text:**Problem Statement:** Find the total area of the shaded region. **Graph Explanation:** The graph is on a coordinate plane with the x-axis and y-axis labeled. The grid consists of squares for precise measurement. The shaded region is between three curves: 1. **Line \( y = x \):** - This is a straight line inclining upward from the origin at a 45-degree angle, represented in blue. 2. **Line \( y = 12 \):** - This is a horizontal line cutting across the graph at \( y = 12 \). 3. **Curve \( y = \frac{x^2}{48} \):** - This is a parabola with its vertex at the origin, opening upwards, represented in red. **Boundary Intersections:** - The line \( y = x \) and the parabola \( y = \frac{x^2}{48} \) intersect at \( x = 0 \). - The line \( y = 12 \) intersects \( y = x \) at \( (12, 12) \). - The parabola \( y = \frac{x^2}{48} \) intersects \( y = 12 \) at \( (24, 12) \). **Shaded Region:** The shaded area is bounded by these segments: - From \( x = 0 \) to \( x = 12 \), the area between the line \( y = x \) and \( y = 12 \). - From \( x = 12 \) to \( x = 24 \), the area between the parabola \( y = \frac{x^2}{48} \) and \( y = 12 \). To find the total area, calculate the area under \( y = 12 \) and subtract the areas under both \( y = x \) and \( y = \frac{x^2}{48} \) within their respective intervals.
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