Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.y = x − 1, y = 0, x = 8; about the x-axis.
I assume you have sketched the region and found it to be a triangle with vertices at
(1,0), (8,0),(8,7)
So, consider the volume as a stack of thin discs of thickness dx. Then
v = ∫[1,8] πr^2 dx
where r = y = x-1
v = ∫[1,8] π(x-1)^2 dx = 343π/3
Or, consider it as a set of nested shells, of thickness dy. Then
v = ∫[0,7] 2πrh dy
where r=y and h = 8-x = 8-(y+1) = 7-y
v = ∫[0,7] 2πy(7-y) dy = 343π/3