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Find the area of the shaded region. y y=8x-r 15 (6, 12) 10 A y=2x 2 4 6 Step 1 To find the area between these curves, we find the integral of the "top function" minus the "bottom function" over the region, which would be the function y = minus the function y =

Find the area of the shaded region. y y=8x-r 15 (6, 12) 10 A y=2x 2 4 6 Step 1 To find the area between these curves, we find the integral of the "top function" minus the "bottom function" over the region, which would be the function y = minus the function y =

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 of the Shaded Region Between Curves** In this example, we aim to find the area of the shaded region bounded by two curves. The given curves in the graph are \( y = 8x - x^2 \) (a downward-opening parabola, in red) and \( y = 2x \) (a straight line, in blue). ### Graph Description - The vertical axis of the graph is labeled \( y \), and the horizontal axis is labeled \( x \). - The intersection points of the curves are crucial as they determine the bounds for integration. In this graph, one such point is clearly marked at \( (6, 12) \). - The shaded region lies between these two curves from \( x = 0 \) to \( x = 6 \). The curves intersect again at another point, but it's not annotated in this graph. ### Step-by-Step Solution **Step 1: Determine the Functions** To find the area between these curves, you need to compute the definite integral of the "top" function minus the "bottom" function over the region they enclose. - The "top" function (the parabola) is: \( y = 8x - x^2 \) - The "bottom" function (the straight line) is: \( y = 2x \) Thus, the integral to find the area of the shaded region is: \[ \int (8x - x^2 - 2x) \, dx \] ### Interactive Prompt To help students better understand, there is an interactive prompt: - Find the integral function by filling in the blanks: \[ y = \boxed{} - \text{the function} \, y = 2x \]
Transcribed Image Text:**Finding the Area of the Shaded Region Between Curves** In this example, we aim to find the area of the shaded region bounded by two curves. The given curves in the graph are \( y = 8x - x^2 \) (a downward-opening parabola, in red) and \( y = 2x \) (a straight line, in blue). ### Graph Description - The vertical axis of the graph is labeled \( y \), and the horizontal axis is labeled \( x \). - The intersection points of the curves are crucial as they determine the bounds for integration. In this graph, one such point is clearly marked at \( (6, 12) \). - The shaded region lies between these two curves from \( x = 0 \) to \( x = 6 \). The curves intersect again at another point, but it's not annotated in this graph. ### Step-by-Step Solution **Step 1: Determine the Functions** To find the area between these curves, you need to compute the definite integral of the "top" function minus the "bottom" function over the region they enclose. - The "top" function (the parabola) is: \( y = 8x - x^2 \) - The "bottom" function (the straight line) is: \( y = 2x \) Thus, the integral to find the area of the shaded region is: \[ \int (8x - x^2 - 2x) \, dx \] ### Interactive Prompt To help students better understand, there is an interactive prompt: - Find the integral function by filling in the blanks: \[ y = \boxed{} - \text{the function} \, y = 2x \]
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