**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.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.2: Graphs Of Equations

Problem 3E

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