The volume of the solid generated by revolving the region bounded by the curves y=4x-x² and y=x around the y-axis and the line x=3 can be calculated using the shell method. The correct integrals are 2π∫ (0 to 3) x(4x-x² - x) dx for the y-axis and 2π∫ (0 to 3) (3-x)(4x-x² - x) dx for the line x=3. Both integrals require proper subtraction of the linear function y=x from the quadratic function y=4x-x². The setup of these integrals is confirmed to be accurate for finding the desired volume.
PREREQUISITESStudents studying calculus, particularly those focusing on volume calculations of solids of revolution, as well as educators teaching integral calculus concepts.
whatlifeforme said:Homework Statement
Find the volume of the solid generated by revolving the region bounded by y=4x-x^2 and y=x about the y-axis and about the line x=3.
Homework Equations
1. y=4x-x^2 and y=x
2. y=4x-x^2 and x=3
The Attempt at a Solution
1. 2∏∫ (0 to 3) (x)(4x-x^2 -x) dx
2. 2∏∫ (0 to 3) (3-x)(4x-x^2 -x) dx