1. Let R be the region in the first quadrant enclosed by the graphs of y=4-X , y=3x , and the y-axis.

Question

a. Find the area of region R.
b. Find the volume of the solid formed by revolving the region R about the x-axis.

Answer 1

need the intersection ....
3x = 4-x
4x=4
x = 1
height of revolved region = radius of rotation
= 4-x - 3x = 4 - 4x

area = ∫(4-4x) dx from 0 to 1
= [4x - 2x^2] from 0 to 1
= (4 - 2) - 0
= 2

volume = π∫ (4-4x) dx from 0 to 1
= π∫(16 - 32x + 16x^2) dx from 0 to 1
= π[16x - 16x^2 + (16/3)x^3 ] from 0 to 1
= π( 16 - 16 + 16/3 - 0)
= 16π/3

check my arithmetic