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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![**Problem Statement:**
Find the area enclosed by the curves \( f(x) = 33 - x^2 \) and \( g(x) = x^2 \).
The area (rounded to two decimal places) is [ ].
**Explanation:**
Given two functions \( f(x) = 33 - x^2 \) and \( g(x) = x^2 \), we are to find the area enclosed by these curves.
1. **Intersection Points:**
To find the limits of integration, first determine where the curves intersect.
Set \( f(x) = g(x) \):
\[
33 - x^2 = x^2
\]
\[
33 = 2x^2
\]
\[
x^2 = \frac{33}{2}
\]
\[
x = \pm \sqrt{\frac{33}{2}}
\]
2. **Integration to Find Area:**
The area between two curves \( f(x) \) and \( g(x) \) from \( x = a \) to \( x = b \) is determined by:
\[
\text{Area} = \int_{a}^{b} [f(x) - g(x)] \, dx
\]
For our functions:
\[
\text{Area} = \int_{-\sqrt{\frac{33}{2}}}^{\sqrt{\frac{33}{2}}} [(33 - x^2) - x^2] \, dx
\]
\[
= \int_{-\sqrt{\frac{33}{2}}}^{\sqrt{\frac{33}{2}}} (33 - 2x^2) \, dx
\]
3. **Compute the Definite Integral:**
\[
\text{Area} = \int_{-\sqrt{\frac{33}{2}}}^{\sqrt{\frac{33}{2}}} 33 \, dx - \int_{-\sqrt{\frac{33}{2}}}^{\sqrt{\frac{33}{2}}} 2x^2 \, dx
\]
Evaluate each integral separately:
\[
\int_{-\sqrt{\frac{33}{2}}}^{\sqrt{\frac{33}{2}}} 33 \, dx](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd379b166-bb3f-49cc-8f67-a1e231fb014b%2F36a3794e-a0d4-4fa5-ba02-5b828bf611c7%2F5xgeg6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the area enclosed by the curves \( f(x) = 33 - x^2 \) and \( g(x) = x^2 \).
The area (rounded to two decimal places) is [ ].
**Explanation:**
Given two functions \( f(x) = 33 - x^2 \) and \( g(x) = x^2 \), we are to find the area enclosed by these curves.
1. **Intersection Points:**
To find the limits of integration, first determine where the curves intersect.
Set \( f(x) = g(x) \):
\[
33 - x^2 = x^2
\]
\[
33 = 2x^2
\]
\[
x^2 = \frac{33}{2}
\]
\[
x = \pm \sqrt{\frac{33}{2}}
\]
2. **Integration to Find Area:**
The area between two curves \( f(x) \) and \( g(x) \) from \( x = a \) to \( x = b \) is determined by:
\[
\text{Area} = \int_{a}^{b} [f(x) - g(x)] \, dx
\]
For our functions:
\[
\text{Area} = \int_{-\sqrt{\frac{33}{2}}}^{\sqrt{\frac{33}{2}}} [(33 - x^2) - x^2] \, dx
\]
\[
= \int_{-\sqrt{\frac{33}{2}}}^{\sqrt{\frac{33}{2}}} (33 - 2x^2) \, dx
\]
3. **Compute the Definite Integral:**
\[
\text{Area} = \int_{-\sqrt{\frac{33}{2}}}^{\sqrt{\frac{33}{2}}} 33 \, dx - \int_{-\sqrt{\frac{33}{2}}}^{\sqrt{\frac{33}{2}}} 2x^2 \, dx
\]
Evaluate each integral separately:
\[
\int_{-\sqrt{\frac{33}{2}}}^{\sqrt{\frac{33}{2}}} 33 \, dx
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