Evaluate the integral f(x) dx, where (2 - x', x < 1 x?, f(x) = x > 1 (Use symbolic notation and fractions where needed.) f(x) dx =

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...
icon
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.
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
steps

Step by step

Solved in 2 steps with 1 images

Blurred answer
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
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: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning