The volume of the solid generated by rotating the region bounded by the given curves about specific axis by any method.
i. If x is the radius of the typical shell, then the circumference =2πx and the height is y
ii. By the shell method, the volume of the solid by rotating the region under the curve y=f(x) about line y=l from a to b is
where 0≤a≤b and g(y)=x
The region bounded by x=y-32 , x=4 and rotated about the line y=1
As region is bounded by x=y-32, x=4 and rotated about the line y=1
Using the shell method,
Radius is (y-1) the circumference is 2π(y-1) and the height is 4-y-32
From the graph, y=1 and y=5 are the y coordinates of the intersection of these two curves.
Therefore, a=1 and b=5
So, the total volume is
V=2π ∫15(-y3+7y2-11y+5 )dy
The volume of the solid obtained by rotating the region bounded by the given curves is