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
![**Evaluate the integral**
\[
\int_{-1}^{2} f(x) \, dx,
\]
**where**
\( f(x) = \begin{cases}
2 - x^3, & x \leq 1 \\
x^2, & x > 1
\end{cases} \)
(Use symbolic notation and fractions where needed.)
\[
\int_{-1}^{2} f(x) \, dx = \underline{\hspace{10cm}}
\]
**Explanation:**
You have a piecewise function \( f(x) \) that is defined differently depending on the value of \( x \):
- For \( x \) less than or equal to 1, \( f(x) = 2 - x^3 \).
- For \( x \) greater than 1, \( f(x) = x^2 \).
To evaluate the definite integral from \(-1\) to \(2\), you need to split the integral at the point where the function's expression changes, which is at \( x = 1 \).
Thus, the integral can be separated into two parts:
1. \(\int_{-1}^{1} (2 - x^3) \, dx\)
2. \(\int_{1}^{2} x^2 \, dx\)
Calculate each integral separately and sum the results.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffafdbcbf-2d89-4a3a-91e5-ac9b5190be04%2Fd6f4a97d-7103-4f30-88e2-c484d030379d%2Ffjsg9r6_processed.png&w=3840&q=75)
Transcribed Image Text:**Evaluate the integral**
\[
\int_{-1}^{2} f(x) \, dx,
\]
**where**
\( f(x) = \begin{cases}
2 - x^3, & x \leq 1 \\
x^2, & x > 1
\end{cases} \)
(Use symbolic notation and fractions where needed.)
\[
\int_{-1}^{2} f(x) \, dx = \underline{\hspace{10cm}}
\]
**Explanation:**
You have a piecewise function \( f(x) \) that is defined differently depending on the value of \( x \):
- For \( x \) less than or equal to 1, \( f(x) = 2 - x^3 \).
- For \( x \) greater than 1, \( f(x) = x^2 \).
To evaluate the definite integral from \(-1\) to \(2\), you need to split the integral at the point where the function's expression changes, which is at \( x = 1 \).
Thus, the integral can be separated into two parts:
1. \(\int_{-1}^{1} (2 - x^3) \, dx\)
2. \(\int_{1}^{2} x^2 \, dx\)
Calculate each integral separately and sum the results.
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 1 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