calculus

1. Let R be the region in the first quadrant enclosed by the graphs of y=4-X , y=3x , and the y-axis.
a. Find the area of region R.
b. Find the volume of the solid formed by revolving the region R about the x-axis.

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  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

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