Question

Find the volume of the solid obtained by revolving the region x=(y-2)^2, the x-axis, the y-axis, about the x-axis.

So I understand that you're supposed to use disc method; however, idk where I evaluated wrong. My final answer is 136π/3, but according to Wolfram Alpha, the answer is 8π/3 and they used shell. Below is my work:

x=(0-2)^2
x=(-2)^2
x=4

x=(y-2)^2
x^(1/2)=y-2
x^(1/2)+2=y

π∫(x^(1/2)+2)^2 dx [0,4]
π∫(x+4(x)^(1/2)+4) dx [0,4]
π[(x^2/2)+(8/3)x^(3/2)+4x] [0,4]
=136π/3

Answer 1

using discs of thickness dx,
v = ∫[0,4] πr^2 dx
where r = y = 2-√x
The parabola has two branches, and you took the upper branch 2+√x, but we want the curve bounded by the axes, which is 2-√x
v = ∫[0,4] π(2-√x)^2 dx

Answer 2

Make your sketch.
x = (y-2)^2
±√x = y - 2
y = 2 + √x or y = 2 - √x
the region that you are revolving has as its boundary y = 2 - √x
and crosses the x-axis at (4,0)
so we want:
π∫ y^2 dx from x=0 to x=4
= π∫ (2-√x)^2 dx from 0 to 4 = π∫ (2-x^(1/2))^2 dx from 0 to 4
= π∫ (4 - 4x^(1/2) + x) dx from 0 to 4
= π [4x - (8/3)x^(3/2) + (1/2)x^2 ] from 0 to 4
= π( 8 -(8/3)(8) + 8 - 0 )
= 8π/3 , as required