# Calculating the Area Between Two Curves## Problem StatementCalculate 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 GraphThe 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 CurvesTo 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 AreaTo 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:**\[ A = \boxed{} \]

Answered: Image /qna-images/answer/dce371f6-d202-4678-b93f-058141046f67.jpg

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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Concept explainers

Cylinders

A cylinder is a three-dimensional solid shape with two parallel and congruent circular bases, joined by a curved surface at a fixed distance. A cylinder has an infinite curvilinear surface.

Cones

A cone is a three-dimensional solid shape having a flat base and a pointed edge at the top. The flat base of the cone tapers smoothly to form the pointed edge known as the apex. The flat base of the cone can either be circular or elliptical. A cone is drawn by joining the apex to all points on the base, using segments, lines, or half-lines, provided that the apex and the base both are in different planes.

Question

# 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:**

\[ A = \boxed{} \]

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