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 a limit equal to
\[
\int_{3}^{7} (x^2 - 8) \, dx
\]
**Explanation:**
The integral given is a definite integral, which calculates the area under the curve of the function \(f(x) = x^2 - 8\) from \(x = 3\) to \(x = 7\). To evaluate this integral, we should find the antiderivative of the function \(f(x)\) and then apply the Fundamental Theorem of Calculus.
1. **Antiderivative:**
The function \(f(x) = x^2 - 8\) can be integrated to find its antiderivative:
\[
\int (x^2 - 8) \, dx = \frac{x^3}{3} - 8x + C
\]
2. **Evaluate from 3 to 7:**
Using the limits of integration, we calculate:
\[
\left[ \frac{x^3}{3} - 8x \right]_{3}^{7} = \left( \frac{7^3}{3} - 8 \times 7 \right) - \left( \frac{3^3}{3} - 8 \times 3 \right)
\]
3. **Solve:**
Calculate each part separately:
- For \(x = 7\):
\[
\frac{7^3}{3} - 8 \times 7 = \frac{343}{3} - 56
\]
- For \(x = 3\):
\[
\frac{3^3}{3} - 8 \times 3 = \frac{27}{3} - 24
\]
Subtract the second result from the first to find the value of the definite integral.
This process will yield the limit that is equal to the value of the definite integral.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbf7825c6-82c4-499b-b5e4-e34c7ca5bb51%2Fd8983eb4-6301-48d0-921a-f1bbaf30632d%2Fs5m2orp_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find a limit equal to
\[
\int_{3}^{7} (x^2 - 8) \, dx
\]
**Explanation:**
The integral given is a definite integral, which calculates the area under the curve of the function \(f(x) = x^2 - 8\) from \(x = 3\) to \(x = 7\). To evaluate this integral, we should find the antiderivative of the function \(f(x)\) and then apply the Fundamental Theorem of Calculus.
1. **Antiderivative:**
The function \(f(x) = x^2 - 8\) can be integrated to find its antiderivative:
\[
\int (x^2 - 8) \, dx = \frac{x^3}{3} - 8x + C
\]
2. **Evaluate from 3 to 7:**
Using the limits of integration, we calculate:
\[
\left[ \frac{x^3}{3} - 8x \right]_{3}^{7} = \left( \frac{7^3}{3} - 8 \times 7 \right) - \left( \frac{3^3}{3} - 8 \times 3 \right)
\]
3. **Solve:**
Calculate each part separately:
- For \(x = 7\):
\[
\frac{7^3}{3} - 8 \times 7 = \frac{343}{3} - 56
\]
- For \(x = 3\):
\[
\frac{3^3}{3} - 8 \times 3 = \frac{27}{3} - 24
\]
Subtract the second result from the first to find the value of the definite integral.
This process will yield the limit that is equal to the value of the definite integral.
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