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
![### 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} -
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