Find the volume of the solid generated by revolving the region about the given line.?

Question

The region in the first quadrant bounded above by the line y = 3, below by the line y = 3x/4, and on the left by the y-axis, about the line y = 3

a. 16π b. 12π c. 6π d. 84π
please help me.

Answer 1

First find the point where y = 3x/4 intersects the line y = 3.

Set the two equations equal.

3x/4 = 3
3x = 12
x = 4

So we will be integrating from x = 0 to x = 4.

Use the disk method.

V = ∫πr²dx = π∫(3 - 3x/4)²dx = π∫(9 - 9x/2 + 9x²/16)dx

= π(9x - 9x²/4 + 3x³/16) | [Evaluated from 0 to 4]

= π(9*4 - 9*16/4 + 3*64/16) - 0

= π(36 - 36 + 12) = 12π

The answer is b. 12π.

Answer 2

v
= ∫π(3-y)^2 dx, x from 0 to 4
= ∫π(3-3x/4)^2 dx, x from 0 to 4
= 12π