find the volume of the solid obtained by rotating the region y=x, y=0, x=4 , x=7 about x=1

Answer 1

The region noted is a rectangle topped by a triangle.

The x centroid for this region is 5.6364 and the total area for the region is 16.5.

The volume of revolution for this region about the vertical line x = 1 is

V = 2*pi*xbar*A where pi = 3.14159, xbar = distance to the centroid from the line x = 1 = 4.6364, &

A = Area = 16.5 So,

V = 2*3.14159*4.6364*16.5 = 480.7

Answer 2

First, draw a diagram of the region.

Using the method of cylindrical shells, take a vertical cross section of the region at a distance of x from the y-axis. Rotate the cross section about the line x = 1 to get a shell of height x, and radius x - 1.

The volume of the shell is 2πx(x-1)dx = 2π(x² - x)dx

Volume of solid = 2π∫_{(from 4 to 7)}(x² - x)dx = 2π[ (1/3)x³ - (1/2)x²]_{(from 4 to 7)}

= 2π[343/3 - 49/2 - (64/3 - 8] = 480.66