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...
Related questions
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**.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning