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
![### Area Under the Curve Calculation
**Question:**
What is the area under the curve for \( f(x) = 3x^2 + 5 \) between \( x = 1 \) and \( x = 10 \)?
**Explanation:**
To find the area under the curve of a function, you need to compute the definite integral of the function over the given interval. In this case, the function is \( f(x) = 3x^2 + 5 \).
### Steps to Solve:
1. **Find the Integral:**
The indefinite integral of \( f(x) = 3x^2 + 5 \) is:
\[
\int (3x^2 + 5) \, dx = x^3 + 5x + C
\]
where \( C \) is the constant of integration.
2. **Evaluate the Definite Integral:**
To find the area from \( x = 1 \) to \( x = 10 \), evaluate the definite integral:
\[
\int_{1}^{10} (3x^2 + 5) \, dx = [x^3 + 5x]_{1}^{10}
\]
3. **Calculate the Result:**
\[
\text{Calculate } (10^3 + 5 \times 10) - (1^3 + 5 \times 1)
\]
### Result:
- Substitute \( x = 10 \): \( 10^3 + 5 \times 10 = 1000 + 50 = 1050 \)
- Substitute \( x = 1 \): \( 1^3 + 5 \times 1 = 1 + 5 = 6 \)
- Resulting Area: \( 1050 - 6 = 1044 \)
Thus, the area under the curve between \( x = 1 \) and \( x = 10 \) is **1044**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66da7bdd-95f3-4b5b-a977-40baa95a9996%2F9b505312-2fe1-481b-b0dd-1dc66248b326%2F56g7iim_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Area Under the Curve Calculation
**Question:**
What is the area under the curve for \( f(x) = 3x^2 + 5 \) between \( x = 1 \) and \( x = 10 \)?
**Explanation:**
To find the area under the curve of a function, you need to compute the definite integral of the function over the given interval. In this case, the function is \( f(x) = 3x^2 + 5 \).
### Steps to Solve:
1. **Find the Integral:**
The indefinite integral of \( f(x) = 3x^2 + 5 \) is:
\[
\int (3x^2 + 5) \, dx = x^3 + 5x + C
\]
where \( C \) is the constant of integration.
2. **Evaluate the Definite Integral:**
To find the area from \( x = 1 \) to \( x = 10 \), evaluate the definite integral:
\[
\int_{1}^{10} (3x^2 + 5) \, dx = [x^3 + 5x]_{1}^{10}
\]
3. **Calculate the Result:**
\[
\text{Calculate } (10^3 + 5 \times 10) - (1^3 + 5 \times 1)
\]
### Result:
- Substitute \( x = 10 \): \( 10^3 + 5 \times 10 = 1000 + 50 = 1050 \)
- Substitute \( x = 1 \): \( 1^3 + 5 \times 1 = 1 + 5 = 6 \)
- Resulting Area: \( 1050 - 6 = 1044 \)
Thus, the area under the curve between \( x = 1 \) and \( x = 10 \) is **1044**.
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