Question

Use the method of cylindrical shells to write out an integral formula for the volume of the solid generated by rotating the region bounded by the curve y = 2x - x^2 and the line y = x about the y-axis.

Answer 1

The answer is "
`(5\pi)/(6)`"

Explanation:

Please find the graph file.

`h= y=2x-x^2\\\\r= x\\\\Area=2\pi* r* h\\\\= 2 \pi * x * (2x-x^2)\\\\= 2 \pi * 2x^2-x^3\\\\volume \ V(x)=\int \ A(x)\ dx\\\\= \int^(x=1)_(x=0) 2\pi (2x^2-x^3)\ dx\\\\= 2\pi [((2x^3)/(3)-(x^4)/(4))]^(1)_(0) \\\\= 2\pi [((2)/(3)-(1)/(4))-(0-0)] \\\\= 2\pi * (5)/(12)\\\\=(5\pi)/(6)\\\\`