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## Homework Statement

Determine the volume of a sold obtained by rotating the region bounded by y=x^2-2x and y=x about the line y =4

## Homework Equations

## The Attempt at a Solution

Let:

y1 = x

y2 = x^2 - 2x.

v be the volume.

y1 and y2 meet where:

x^2 - 2x = x

x^2 - 3x = 0

x(x - 3) = 0

x = 0, x = 3.

The straight line is closer to y = 4 than the parabola between x = 0 and x = 3.

v = pi int(0, 3)[ (y2 - 4)^2 - (y1 - 4)^2 ] dx

= pi int(0, 3) [ (x^2 - 2x - 4)^2 - (x - 4)^2 ] dx

= pi int(0, 3) [ x^4 + 4x^2 + 16 - 4x^3 + 16x - 8x^2 - x^2 - 16 + 8x ] dx

= pi int(0, 3) [ x^4 - 4x^3 - 5x^2 + 24x ] dx

= pi [ x^5 / 5 - x^4 - 5x^3 / 3 + 12x^2 ](0, 3)

= pi [ 3^5 / 5 - 3^4 - 5 * 3^3 / 3 + 12 * 3^2 ]

= 30.6 pi.

IS THIS RIGHT?