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
![### Integral Problem Solving
**Problem Statement:**
2. Write an integral that represents the shaded region on the graph, and then evaluate your integral.
**Graph and Function:**
The graph presented depicts a quadratic function, \( f(x) = x^2 - 2x \), which describes a parabola opening upwards.
**Steps and Explanation:**
1. **Identifying the Bounded Region:**
The shaded region under the curve of the function \( f(x) = x^2 - 2x \) needs to be determined. To identify the points that bound this region, we observe the points of intersection where the curve crosses the x-axis.
2. **Finding Intersection Points:**
Solve \( x^2 - 2x = 0 \).
\[
x(x - 2) = 0
\]
This gives us \( x = 0 \) and \( x = 2 \).
3. **Setting Up the Integral:**
The integral will be set up from \( x = 0 \) to \( x = 2 \), which are the points of intersection. The integral that represents the shaded region is:
\[
\int_{0}^{2} (x^2 - 2x) \, dx
\]
4. **Evaluating the Integral:**
To evaluate the integral:
\[
\int (x^2 - 2x) \, dx = \int x^2 \, dx - \int 2x \, dx
\]
Compute each term separately:
\[
\int x^2 \, dx = \frac{x^3}{3} + C
\]
\[
\int 2x \, dx = x^2 + C
\]
Combine and apply the limits from 0 to 2:
\[
\left[ \frac{x^3}{3} - x^2 \right]_{0}^{2}
\]
Substitute the upper and lower bounds:
\[
\left( \frac{2^3}{3} - 2^2 \right) - \left( \frac{0^3}{3} - 0^2 \right)
\]
Simplify:
\[
\left( \frac{8}{3} -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8a5ae68f-9b8c-40fd-acc4-5e8548032b37%2F828ae9cc-5fd3-4b8d-8d9e-5eaaa00108ea%2Fmgjgra_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Integral Problem Solving
**Problem Statement:**
2. Write an integral that represents the shaded region on the graph, and then evaluate your integral.
**Graph and Function:**
The graph presented depicts a quadratic function, \( f(x) = x^2 - 2x \), which describes a parabola opening upwards.
**Steps and Explanation:**
1. **Identifying the Bounded Region:**
The shaded region under the curve of the function \( f(x) = x^2 - 2x \) needs to be determined. To identify the points that bound this region, we observe the points of intersection where the curve crosses the x-axis.
2. **Finding Intersection Points:**
Solve \( x^2 - 2x = 0 \).
\[
x(x - 2) = 0
\]
This gives us \( x = 0 \) and \( x = 2 \).
3. **Setting Up the Integral:**
The integral will be set up from \( x = 0 \) to \( x = 2 \), which are the points of intersection. The integral that represents the shaded region is:
\[
\int_{0}^{2} (x^2 - 2x) \, dx
\]
4. **Evaluating the Integral:**
To evaluate the integral:
\[
\int (x^2 - 2x) \, dx = \int x^2 \, dx - \int 2x \, dx
\]
Compute each term separately:
\[
\int x^2 \, dx = \frac{x^3}{3} + C
\]
\[
\int 2x \, dx = x^2 + C
\]
Combine and apply the limits from 0 to 2:
\[
\left[ \frac{x^3}{3} - x^2 \right]_{0}^{2}
\]
Substitute the upper and lower bounds:
\[
\left( \frac{2^3}{3} - 2^2 \right) - \left( \frac{0^3}{3} - 0^2 \right)
\]
Simplify:
\[
\left( \frac{8}{3} -
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 3 steps with 4 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.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