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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![### Checkpoint 1.24
**Problem Statement:**
Integrate the function
\[ \int_{-2}^{2} x^4 \, dx. \]
---
This content will appear as a part of an educational website focused on teaching integral calculus concepts and problem-solving techniques.
**Explanation:**
The given problem asks for the definite integral of \( x^4 \) from \(-2\) to \(2\). In integral calculus, this can be solved using the power rule of integration.
The power rule of integration states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
where \( n \) is any real number except \(-1\), and \( C \) is the constant of integration.
For this problem, apply the power rule to integrate \( x^4 \), and then evaluate the definite integral from \(-2\) to \(2\).
**Solution:**
1. **Integrate \( x^4 \):**
\[ \int x^4 \, dx = \frac{x^{5}}{5} + C \]
2. **Evaluate the definite integral from -2 to 2:**
\[ \left. \frac{x^{5}}{5} \right|_{-2}^{2} = \left( \frac{2^{5}}{5} \right) - \left( \frac{(-2)^{5}}{5} \right) \]
3. **Simplify the expression:**
\[ = \frac{32}{5} - \frac{-32}{5} \]
\[ = \frac{32}{5} + \frac{32}{5} \]
\[ = \frac{64}{5} \]
Thus, the integral \( \int_{-2}^{2} x^4 \, dx \) evaluates to \(\frac{64}{5}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11c3e735-47a2-4be2-bcf0-76944b0744ea%2Fed1d6713-e7a0-48d7-b4d5-8ee4c4b47b0c%2F1tk7u1tk_processed.png&w=3840&q=75)
Transcribed Image Text:### Checkpoint 1.24
**Problem Statement:**
Integrate the function
\[ \int_{-2}^{2} x^4 \, dx. \]
---
This content will appear as a part of an educational website focused on teaching integral calculus concepts and problem-solving techniques.
**Explanation:**
The given problem asks for the definite integral of \( x^4 \) from \(-2\) to \(2\). In integral calculus, this can be solved using the power rule of integration.
The power rule of integration states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
where \( n \) is any real number except \(-1\), and \( C \) is the constant of integration.
For this problem, apply the power rule to integrate \( x^4 \), and then evaluate the definite integral from \(-2\) to \(2\).
**Solution:**
1. **Integrate \( x^4 \):**
\[ \int x^4 \, dx = \frac{x^{5}}{5} + C \]
2. **Evaluate the definite integral from -2 to 2:**
\[ \left. \frac{x^{5}}{5} \right|_{-2}^{2} = \left( \frac{2^{5}}{5} \right) - \left( \frac{(-2)^{5}}{5} \right) \]
3. **Simplify the expression:**
\[ = \frac{32}{5} - \frac{-32}{5} \]
\[ = \frac{32}{5} + \frac{32}{5} \]
\[ = \frac{64}{5} \]
Thus, the integral \( \int_{-2}^{2} x^4 \, dx \) evaluates to \(\frac{64}{5}\).
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