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
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:
\[ y = \boxed{} - \text{the function} \, y = 2x \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2e97f32-5988-4deb-9547-eb6ce37eb1f3%2F9f6afa1a-fb69-4000-b667-1f05cb0a9986%2F1ymj61_processed.png&w=3840&q=75)
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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