### Finding the Area of an Enclosed Region Using IntegrationTo find the area of the region enclosed by the equations $ y = 4.75x $ and $ x = 8.5 - y^2 $, follow these steps. The graph provides a visual guide for the enclosed region.#### Graph Explanation:The graph displays a region bounded by two curves:1. $ y = 4.75x $, which is a straight line.2. $ x = 8.5 - y^2 $, which is a parabola opening to the left.#### Steps to Find the Enclosed Area:1. **Use Horizontal Strips to Find the Area**: Integrate with respect to $ y $ to find the area of the region.2. **Find the $ y $-coordinates of Intersection**: Determine the points where the line $ y = 4.75x $ intersects the parabola $ x = 8.5 - y^2 $. Solve the system of equations: \[ y = 4.75x \] and \[ x = 8.5 - y^2 \] Substitute $ y $ from the first equation into the second equation: \[ x = 8.5 - (4.75x)^2 \] Simplify to find the $ y $-values at the points of intersection. These values will be the limits of integration. - Lower limit $ c = $ [Enter Value] - Upper limit $ d = $ [Enter Value]3. **Set Up the Integral**: To find the area of the enclosed region from $ c $ to $ d $: \[ \int_{c}^{d} g(y) \, dy \] Here, $ g(y) $ is the difference of the functions representing the curves, rearranged in terms of $ y $ if necessary.4. **Evaluate the Integral**: Solve the definite integral to find the area. - [Set Up Integral] - Evaluate the definite integral to find the area $ \text{Area} = $ [Enter Value]### Example Calculation(For simplicity, assume example values for the coordinates and corresponding integral setup).1. Determine the intersection points: - Solve for \( y \

Given two equations To find the intersection of both equations, use substitution method

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Find the area of the region enclosed by y = 4.75x and r = 8.5 – y?. 5 6 7 8 on

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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Question

### Finding the Area of an Enclosed Region Using Integration

To find the area of the region enclosed by the equations $ y = 4.75x $ and $ x = 8.5 - y^2 $, follow these steps. The graph provides a visual guide for the enclosed region.

#### Graph Explanation:
The graph displays a region bounded by two curves:
1. $ y = 4.75x $, which is a straight line.
2. $ x = 8.5 - y^2 $, which is a parabola opening to the left.

#### Steps to Find the Enclosed Area:

1. **Use Horizontal Strips to Find the Area**:
Integrate with respect to $ y $ to find the area of the region.

2. **Find the $ y $-coordinates of Intersection**:
Determine the points where the line $ y = 4.75x $ intersects the parabola $ x = 8.5 - y^2 $.

Solve the system of equations:

\[
y = 4.75x
\]

and

\[
x = 8.5 - y^2
\]

Substitute $ y $ from the first equation into the second equation:

\[
x = 8.5 - (4.75x)^2
\]

Simplify to find the $ y $-values at the points of intersection. These values will be the limits of integration.

- Lower limit $ c = $ [Enter Value]
- Upper limit $ d = $ [Enter Value]

3. **Set Up the Integral**:
To find the area of the enclosed region from $ c $ to $ d $:

\[
\int_{c}^{d} g(y) \, dy
\]

Here, $ g(y) $ is the difference of the functions representing the curves, rearranged in terms of $ y $ if necessary.

4. **Evaluate the Integral**:
Solve the definite integral to find the area.

- [Set Up Integral]
- Evaluate the definite integral to find the area $ \text{Area} = $ [Enter Value]

### Example Calculation
(For simplicity, assume example values for the coordinates and corresponding integral setup).

1. Determine the intersection points:
- Solve for \( y \

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