Consider the region bounded by the parabola y=x^2 and the line y=16 .

Question

(a) What is the volume of the solid generated when revolving this region about the line y=16 ?

Answer 1

first sketch it. then determine where the y=16 line intersects (-4,16)(4,16)

Now, lets integrate from the parabola to the line

dArea=dx*(16-y)=(16-x^2)dx

area= INT (16-x^2)dx from x=-4 to 4

area= [16x-1/3 x^3] over the limits

area= 16*4-1/3 4^3 - 16(-4)+1/3 (-4)^3

=2(16*4 - 1/3 4^3)

and you can finish it.

You could have recognized symettry, and just integrated from x=0 to 4, then doubled the area.