Find the area of the region that is bounded above by the curve f(x) = (x + 2)² and the line g(x) = 4 - x and bounded below by the x-axis. Enter an exact answer.
Find the area of the region that is bounded above by the curve f(x) = (x + 2)² and the line g(x) = 4 - x and bounded below by the x-axis. Enter an exact answer.
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
![**Problem Statement:**
Find the area of the region that is bounded above by the curve \( f(x) = (x + 2)^2 \) and the line \( g(x) = 4 - x \) and bounded below by the x-axis.
Enter an exact answer.
---
**Solution Guide:**
1. **Identify the Intersection Points:**
To find the limits of integration, solve for \( x \) where \( f(x) = g(x) \).
\[
(x + 2)^2 = 4 - x
\]
2. **Solve the Equation Step-by-Step:**
Expand \( (x + 2)^2 \):
\[
x^2 + 4x + 4 = 4 - x
\]
Combine like terms:
\[
x^2 + 4x + 4 - 4 + x = 0
\]
Simplify further:
\[
x^2 + 5x = 0
\]
Factorize the quadratic equation:
\[
x(x + 5) = 0
\]
The solutions are:
\[
x = 0 \quad \text{and} \quad x = -5
\]
3. **Set Up the Integral:**
The area is calculated by integrating the difference between \( g(x) \) and \( f(x) \) from \( x = -5 \) to \( x = 0 \):
\[
\text{Area} = \int_{-5}^{0} \left[(4 - x) - (x + 2)^2\right] \, dx
\]
4. **Simplify Inside the Integral:**
\[
\int_{-5}^{0} \left[ 4 - x - (x^2 + 4x + 4) \right] \, dx \\
= \int_{-5}^{0} \left[ 4 - x - x^2 - 4x - 4 \right] \, dx \\
= \int_{-5}^{0} \left[ -x^2 - 5x \right] \, dx
\]
5. **Compute the Integral:**
\[
\int -x^2 -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0ea55b65-6a30-46ce-96b8-2b732a599bb3%2Fa52a754c-15b3-4445-bc11-89b2f876594a%2Flxltgxe_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the area of the region that is bounded above by the curve \( f(x) = (x + 2)^2 \) and the line \( g(x) = 4 - x \) and bounded below by the x-axis.
Enter an exact answer.
---
**Solution Guide:**
1. **Identify the Intersection Points:**
To find the limits of integration, solve for \( x \) where \( f(x) = g(x) \).
\[
(x + 2)^2 = 4 - x
\]
2. **Solve the Equation Step-by-Step:**
Expand \( (x + 2)^2 \):
\[
x^2 + 4x + 4 = 4 - x
\]
Combine like terms:
\[
x^2 + 4x + 4 - 4 + x = 0
\]
Simplify further:
\[
x^2 + 5x = 0
\]
Factorize the quadratic equation:
\[
x(x + 5) = 0
\]
The solutions are:
\[
x = 0 \quad \text{and} \quad x = -5
\]
3. **Set Up the Integral:**
The area is calculated by integrating the difference between \( g(x) \) and \( f(x) \) from \( x = -5 \) to \( x = 0 \):
\[
\text{Area} = \int_{-5}^{0} \left[(4 - x) - (x + 2)^2\right] \, dx
\]
4. **Simplify Inside the Integral:**
\[
\int_{-5}^{0} \left[ 4 - x - (x^2 + 4x + 4) \right] \, dx \\
= \int_{-5}^{0} \left[ 4 - x - x^2 - 4x - 4 \right] \, dx \\
= \int_{-5}^{0} \left[ -x^2 - 5x \right] \, dx
\]
5. **Compute the Integral:**
\[
\int -x^2 -
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