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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![### Calculating Enclosed Area for the Curve \( y^2 = x(1 - x) \)
**Question:**
The area enclosed by the curve \( y^2 = x(1 - x) \) is given by:
1. \( 2 \int_{0}^{1} x \sqrt{1 - x} \, dx \)
2. \( 2 \int_{0}^{1} \sqrt{x - x^2} \, dx \)
3. \( 4 \int_{0}^{1} \sqrt{x - x^2} \, dx \)
4. None of these
**Explanation:**
This question aims to determine the correct integral expression to calculate the area enclosed by the parabola \( y^2 = x(1 - x) \).
### Options Breakdown:
1. **Option 1:**
\[
2 \int_{0}^{1} x \sqrt{1 - x} \, dx
\]
This integral expression multiplies the integrand \( x \sqrt{1 - x} \) by 2 and integrates it from 0 to 1.
2. **Option 2:**
\[
2 \int_{0}^{1} \sqrt{x - x^2} \, dx
\]
This integral expression simplifies the integrand inside the square root to \( \sqrt{x - x^2} \), multiplies by 2, and integrates from 0 to 1.
3. **Option 3:**
\[
4 \int_{0}^{1} \sqrt{x - x^2} \, dx
\]
Similar to Option 2 but with a different constant, this integral expression multiplies the integrand \( \sqrt{x - x^2} \) by 4 and integrates from 0 to 1.
4. **Option 4: None of these**
This option suggests that none of the provided integral expressions accurately represent the area enclosed by the curve \( y^2 = x(1 - x) \).
When solving such problems, understanding the derivation of the integral expressions and verifying the boundaries and constants is crucial. The correct integral must align with the geometric interpretation of the area under the curve.
### Graphical Explanation:
Unfortunately, no graphs or diagrams are provided in the image. However, typically such problems would include plotting both sides](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4bf8d9f8-9f3c-4f11-9f6a-4feeeac00eb3%2F0e97d742-3296-472a-bf82-3e43353f8121%2F65gw49_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating Enclosed Area for the Curve \( y^2 = x(1 - x) \)
**Question:**
The area enclosed by the curve \( y^2 = x(1 - x) \) is given by:
1. \( 2 \int_{0}^{1} x \sqrt{1 - x} \, dx \)
2. \( 2 \int_{0}^{1} \sqrt{x - x^2} \, dx \)
3. \( 4 \int_{0}^{1} \sqrt{x - x^2} \, dx \)
4. None of these
**Explanation:**
This question aims to determine the correct integral expression to calculate the area enclosed by the parabola \( y^2 = x(1 - x) \).
### Options Breakdown:
1. **Option 1:**
\[
2 \int_{0}^{1} x \sqrt{1 - x} \, dx
\]
This integral expression multiplies the integrand \( x \sqrt{1 - x} \) by 2 and integrates it from 0 to 1.
2. **Option 2:**
\[
2 \int_{0}^{1} \sqrt{x - x^2} \, dx
\]
This integral expression simplifies the integrand inside the square root to \( \sqrt{x - x^2} \), multiplies by 2, and integrates from 0 to 1.
3. **Option 3:**
\[
4 \int_{0}^{1} \sqrt{x - x^2} \, dx
\]
Similar to Option 2 but with a different constant, this integral expression multiplies the integrand \( \sqrt{x - x^2} \) by 4 and integrates from 0 to 1.
4. **Option 4: None of these**
This option suggests that none of the provided integral expressions accurately represent the area enclosed by the curve \( y^2 = x(1 - x) \).
When solving such problems, understanding the derivation of the integral expressions and verifying the boundaries and constants is crucial. The correct integral must align with the geometric interpretation of the area under the curve.
### Graphical Explanation:
Unfortunately, no graphs or diagrams are provided in the image. However, typically such problems would include plotting both sides
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