Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Question
100%
Find the area of the region. See Examples 1, 2, 3, and 4.
f(x)  =  x2 − 10x + 9
g(x)  =  −x2 + 8x + 9
The image is a mathematical problem asking to find the area of the region between two curves. The functions are given as: - \( f(x) = x^2 - 10x + 9 \) - \( g(x) = -x^2 + 8x + 9 \) ### Graph Explanation: The graph illustrates two quadratic functions, \( f(x) \) and \( g(x) \), on a coordinate plane with the x-axis ranging from 0 to 10 and the y-axis ranging from -10 to 20. - The function \( f(x) \) is a parabola opening upwards. - The function \( g(x) \) is a parabola opening downwards. The shaded blue region within the graph indicates the area between the two curves, which is enclosed between the intersection points of the curves. ### Steps to Solve: To find the area of the region between these two functions, one would typically: 1. Determine the points of intersection by setting \( f(x) = g(x) \) and solving for \( x \). 2. Integrate the difference \( g(x) - f(x) \) with respect to \( x \) over the interval determined by the intersection points. This process would yield the required area between the curves, accounting for the vertical distance between the two functions over the specified range.
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Transcribed Image Text:The image is a mathematical problem asking to find the area of the region between two curves. The functions are given as: - \( f(x) = x^2 - 10x + 9 \) - \( g(x) = -x^2 + 8x + 9 \) ### Graph Explanation: The graph illustrates two quadratic functions, \( f(x) \) and \( g(x) \), on a coordinate plane with the x-axis ranging from 0 to 10 and the y-axis ranging from -10 to 20. - The function \( f(x) \) is a parabola opening upwards. - The function \( g(x) \) is a parabola opening downwards. The shaded blue region within the graph indicates the area between the two curves, which is enclosed between the intersection points of the curves. ### Steps to Solve: To find the area of the region between these two functions, one would typically: 1. Determine the points of intersection by setting \( f(x) = g(x) \) and solving for \( x \). 2. Integrate the difference \( g(x) - f(x) \) with respect to \( x \) over the interval determined by the intersection points. This process would yield the required area between the curves, accounting for the vertical distance between the two functions over the specified range.
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