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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![**Evaluate the integral:**
\[ \int_{0}^{3} \sqrt{x^2 + 9} \, dx \]
This problem asks you to evaluate the definite integral from 0 to 3 of the function \(\sqrt{x^2 + 9}\). This is a standard integral problem often encountered in calculus, and it involves finding the area under the curve described by the integrand over the given interval.
**Steps to Solve:**
1. **Substitution**: You could use a trigonometric substitution to simplify the integral.
2. **Evaluation**: Once the integral is simplified, integrate using the appropriate techniques, apply the limits, and solve.
This integral can be solved using the trigonometric substitution \( x = 3\tan\theta \), which will simplify the square root to a trigonometric function.
For a detailed step-by-step solution, you may refer to:
1. **Applying Substitution**: Let \( x = 3\tan\theta \), then \( dx = 3\sec^2\theta \, d\theta \).
2. **Changing Limits**: When \( x = 0 \), \( \theta = 0 \); and when \( x = 3 \), \( \theta = \frac{\pi}{4} \).
3. **Rewriting the Integral**: Substitute into the integral and simplify.
4. **Integration**: Perform the integration in terms of \(\theta\).
5. **Back Substitution**: Convert back to \( x \) and apply the limit to find the result.
Understanding how to perform these steps properly is crucial in tackling similar integrals in calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98eabd66-347c-4504-bb01-99b451e932fa%2F1a46b383-1e0e-4698-b9f6-4dd981e34fcf%2Fmfrho4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Evaluate the integral:**
\[ \int_{0}^{3} \sqrt{x^2 + 9} \, dx \]
This problem asks you to evaluate the definite integral from 0 to 3 of the function \(\sqrt{x^2 + 9}\). This is a standard integral problem often encountered in calculus, and it involves finding the area under the curve described by the integrand over the given interval.
**Steps to Solve:**
1. **Substitution**: You could use a trigonometric substitution to simplify the integral.
2. **Evaluation**: Once the integral is simplified, integrate using the appropriate techniques, apply the limits, and solve.
This integral can be solved using the trigonometric substitution \( x = 3\tan\theta \), which will simplify the square root to a trigonometric function.
For a detailed step-by-step solution, you may refer to:
1. **Applying Substitution**: Let \( x = 3\tan\theta \), then \( dx = 3\sec^2\theta \, d\theta \).
2. **Changing Limits**: When \( x = 0 \), \( \theta = 0 \); and when \( x = 3 \), \( \theta = \frac{\pi}{4} \).
3. **Rewriting the Integral**: Substitute into the integral and simplify.
4. **Integration**: Perform the integration in terms of \(\theta\).
5. **Back Substitution**: Convert back to \( x \) and apply the limit to find the result.
Understanding how to perform these steps properly is crucial in tackling similar integrals in calculus.
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