Calculate the area A of the region between the two curves 8- x = y – 2y and x = 2y + 12. 6- (Express your answer in exact form. Use symbolic notation 4 and fractions where needed.) 6 8 10 12 14 16 18 20 22 24 26 28 -8- -10-
Calculate the area A of the region between the two curves 8- x = y – 2y and x = 2y + 12. 6- (Express your answer in exact form. Use symbolic notation 4 and fractions where needed.) 6 8 10 12 14 16 18 20 22 24 26 28 -8- -10-
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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![# Calculating the Area Between Two Curves
## Problem Statement
Calculate the area \( A \) of the region between the two curves \( x = y^2 - 2y \) and \( x = 2y + 12 \).
(Express your answer in exact form. Use symbolic notation and fractions where needed.)
**Equation for Area:**
\[ A = \]
## Explanation of Graph
The graph on the right side shows two curves:
1. **Parabolic Curve (Blue Curve):** Represented by the equation \( x = y^2 - 2y \). This is a parabola that opens to the right.
2. **Linear Line (Red Line):** Represented by the equation \( x = 2y + 12 \). This is a straight line with a positive slope.
### Intersection Points of Curves
To find the area between these curves, we first need to find their points of intersection. Solve the system of equations by equating \( y^2 - 2y \) and \( 2y + 12 \):
\[ y^2 - 2y = 2y + 12 \]
\[ y^2 - 4y - 12 = 0 \]
Solve this quadratic equation to find the values of \( y \).
### Integration for Area
To find the area \( A \) between the curves, integrate the difference between the functions over the interval determined by the points of intersection \( y = a \) to \( y = b \).
The area \( A \) is given by:
\[ A = \int_{a}^{b} \left[ (2y + 12) - (y^2 - 2y) \right] \, dy \]
This integral represents the area between the curves from \( y = a \) to \( y = b \).
Fill in the box provided with the exact value of the area once computed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1f133c63-1716-41d7-8ec5-0f87428512d1%2Fdce371f6-d202-4678-b93f-058141046f67%2Fhkxvpnj_processed.png&w=3840&q=75)
Transcribed Image Text:# Calculating the Area Between Two Curves
## Problem Statement
Calculate the area \( A \) of the region between the two curves \( x = y^2 - 2y \) and \( x = 2y + 12 \).
(Express your answer in exact form. Use symbolic notation and fractions where needed.)
**Equation for Area:**
\[ A = \]
## Explanation of Graph
The graph on the right side shows two curves:
1. **Parabolic Curve (Blue Curve):** Represented by the equation \( x = y^2 - 2y \). This is a parabola that opens to the right.
2. **Linear Line (Red Line):** Represented by the equation \( x = 2y + 12 \). This is a straight line with a positive slope.
### Intersection Points of Curves
To find the area between these curves, we first need to find their points of intersection. Solve the system of equations by equating \( y^2 - 2y \) and \( 2y + 12 \):
\[ y^2 - 2y = 2y + 12 \]
\[ y^2 - 4y - 12 = 0 \]
Solve this quadratic equation to find the values of \( y \).
### Integration for Area
To find the area \( A \) between the curves, integrate the difference between the functions over the interval determined by the points of intersection \( y = a \) to \( y = b \).
The area \( A \) is given by:
\[ A = \int_{a}^{b} \left[ (2y + 12) - (y^2 - 2y) \right] \, dy \]
This integral represents the area between the curves from \( y = a \) to \( y = b \).
Fill in the box provided with the exact value of the area once computed.
![**Find the area of the shaded region bounded by \( y = 4x \) and \( y = x \sqrt{25 - x^2} \) in the figure.**
**(Give an exact answer. Use symbolic notation and fractions where needed.)**
The figure includes a graph where the curve represented by \( y = 4x \) (a straight line) intersects with the curve \( y = x \sqrt{25 - x^2} \) (a part of a circle) forming a bounded shaded area. The x-axis and y-axis are labeled, and the region of interest is shaded in pink where the two curves intersect.
Given these curves:
1. **\( y = 4x \)** (a linear function)
2. **\( y = x \sqrt{25 - x^2} \)** (part of a circle or related to a semi-circle)
The goal is to determine the exact area enclosed between these two curves.
**Solution box:**
\[ A = \boxed{} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1f133c63-1716-41d7-8ec5-0f87428512d1%2Fdce371f6-d202-4678-b93f-058141046f67%2Ft867bx_processed.png&w=3840&q=75)
Transcribed Image Text:**Find the area of the shaded region bounded by \( y = 4x \) and \( y = x \sqrt{25 - x^2} \) in the figure.**
**(Give an exact answer. Use symbolic notation and fractions where needed.)**
The figure includes a graph where the curve represented by \( y = 4x \) (a straight line) intersects with the curve \( y = x \sqrt{25 - x^2} \) (a part of a circle) forming a bounded shaded area. The x-axis and y-axis are labeled, and the region of interest is shaded in pink where the two curves intersect.
Given these curves:
1. **\( y = 4x \)** (a linear function)
2. **\( y = x \sqrt{25 - x^2} \)** (part of a circle or related to a semi-circle)
The goal is to determine the exact area enclosed between these two curves.
**Solution box:**
\[ A = \boxed{} \]
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