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Find the area of the region. See Examples 1, 2, 3, and 4. f(x) +² + 6x + 9 g(x) = 6x + 25 -10 -5 y 50 40 30 20 g 10 f 5 X

Find the area of the region. See Examples 1, 2, 3, and 4. f(x) +² + 6x + 9 g(x) = 6x + 25 -10 -5 y 50 40 30 20 g 10 f 5 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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## Finding the Area Between Curves To determine the area of the shaded region between two functions, examine the given equations and graph. ### Functions - \( f(x) = x^2 + 6x + 9 \): This is a quadratic function represented by a parabola opening upwards. - \( g(x) = 6x + 25 \): This is a linear function represented by a straight line. ### Graph Details - The graph displays two curves: one of the quadratic function \( f(x) \) and one of the linear function \( g(x) \). - The shaded region represents the area between these two curves. ### Observations - The parabola \( f(x) \) intersects the line \( g(x) \) at two points. - The shaded area is bounded by these points of intersection and represents the region where \( g(x) > f(x) \). ### Steps to Find the Area 1. **Find Points of Intersection:** - Set \( f(x) = g(x) \) and solve for \( x \) to find the points of intersection. 2. **Set Up the Integral:** - The area \( A \) between the curves from \( x = a \) to \( x = b \) is given by: \[ A = \int_{a}^{b} [g(x) - f(x)] \,dx \] - Compute the definite integral of the difference between the functions over the interval determined by the points of intersection. By following these steps, you can accurately find the area of the region between the parabola and the line on the graph.
Transcribed Image Text:## Finding the Area Between Curves To determine the area of the shaded region between two functions, examine the given equations and graph. ### Functions - \( f(x) = x^2 + 6x + 9 \): This is a quadratic function represented by a parabola opening upwards. - \( g(x) = 6x + 25 \): This is a linear function represented by a straight line. ### Graph Details - The graph displays two curves: one of the quadratic function \( f(x) \) and one of the linear function \( g(x) \). - The shaded region represents the area between these two curves. ### Observations - The parabola \( f(x) \) intersects the line \( g(x) \) at two points. - The shaded area is bounded by these points of intersection and represents the region where \( g(x) > f(x) \). ### Steps to Find the Area 1. **Find Points of Intersection:** - Set \( f(x) = g(x) \) and solve for \( x \) to find the points of intersection. 2. **Set Up the Integral:** - The area \( A \) between the curves from \( x = a \) to \( x = b \) is given by: \[ A = \int_{a}^{b} [g(x) - f(x)] \,dx \] - Compute the definite integral of the difference between the functions over the interval determined by the points of intersection. By following these steps, you can accurately find the area of the region between the parabola and the line on the graph.
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