**Problem 3:**The region bounded by the curves $ y = x^3 $, $ y = x $, and the line $ x \geq 0 $ is rotated about the x-axis.---This problem involves a classic calculus technique of finding the volume of a solid of revolution. To solve it, you'll typically use the disk or washer method, examining the area of the region bounded by the given curves and the specified axis. ### Key Steps to Solve:1. **Identify the Intersection Points:** - Set $ y = x^3 $ equal to $ y = x $ and solve for $ x $. - This gives you the points of intersection which are crucial for determining the limits of integration.2. **Determine the Volume:** - Use the disk method if the bounded region is being revolved around the x-axis: \[ V = \pi \int [R(x)^2 - r(x)^2] \, dx \] Where $ R(x) $ is the outer radius and $ r(x) $ is the inner radius, determined by $ y = x $ and $ y = x^3 $.3. **Calculate the Integral:** - Calculate the definite integral within the limits found from the points of intersection.This problem explores the application of integration techniques to real-world geometric problems, enhancing understanding in calculus and solid geometry contexts. The image contains two graphs side by side:1. **Left Graph**: - The axes are labeled $ x $ and $ y $. - Two curves are plotted: - The line $ y = x $, which is a straight line passing through the origin with a slope of 1. - The curve $ y = x^3 $, which passes through the origin and is concave upwards. - The region between the two curves is shaded. This shaded area is bounded by the curves $ y = x $ and $ y = x^3 $, and is between the points $ (0, 0) $ and $ (1, 1) $.2. **Right Graph**: - The axes are labeled $ z $ and $ x $. - The diagram shows a three-dimensional solid that resembles a horn or trumpet shape. - This solid is a result of rotating the shaded area between $ y = x $ and $ y = x^3 $ in the left graph around the $ x $-axis.In the top left corner, there is a numerical expression written in red: $ .4\pi/21 $.

Answered: Image /qna-images/answer/7feaa975-a438-4344-994e-2b6bab379a80.jpg

Answered: 3. The region bounded by y = x³, y = x,…

Math

Calculus

3. The region bounded by y = x³, y = x, x > 0 is rotated about the x – - axis.

Holt Mcdougal Larson Pre-algebra: Student Edition 2012

1st Edition

ISBN:9780547587776

Author:HOLT MCDOUGAL

Publisher:HOLT MCDOUGAL

Chapter8: Linear Functions

Section: Chapter Questions

Problem 3CR

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Question

**Problem 3:**

The region bounded by the curves $ y = x^3 $, $ y = x $, and the line $ x \geq 0 $ is rotated about the x-axis.

---

This problem involves a classic calculus technique of finding the volume of a solid of revolution. To solve it, you'll typically use the disk or washer method, examining the area of the region bounded by the given curves and the specified axis.

### Key Steps to Solve:
1. **Identify the Intersection Points:**
- Set $ y = x^3 $ equal to $ y = x $ and solve for $ x $.
- This gives you the points of intersection which are crucial for determining the limits of integration.

2. **Determine the Volume:**
- Use the disk method if the bounded region is being revolved around the x-axis:
\[
V = \pi \int [R(x)^2 - r(x)^2] \, dx
\]
Where $ R(x) $ is the outer radius and $ r(x) $ is the inner radius, determined by $ y = x $ and $ y = x^3 $.

3. **Calculate the Integral:**
- Calculate the definite integral within the limits found from the points of intersection.

This problem explores the application of integration techniques to real-world geometric problems, enhancing understanding in calculus and solid geometry contexts.

The image contains two graphs side by side:

1. **Left Graph**:
- The axes are labeled $ x $ and $ y $.
- Two curves are plotted:
- The line $ y = x $, which is a straight line passing through the origin with a slope of 1.
- The curve $ y = x^3 $, which passes through the origin and is concave upwards.
- The region between the two curves is shaded. This shaded area is bounded by the curves $ y = x $ and $ y = x^3 $, and is between the points $ (0, 0) $ and $ (1, 1) $.

2. **Right Graph**:
- The axes are labeled $ z $ and $ x $.
- The diagram shows a three-dimensional solid that resembles a horn or trumpet shape.
- This solid is a result of rotating the shaded area between $ y = x $ and $ y = x^3 $ in the left graph around the $ x $-axis.

In the top left corner, there is a numerical expression written in red: $ .4\pi/21 $.

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