How do you find the volume of the solid obtained by rotating the region bounded by the curves #y=2x^2+5#, and #y=x+3# and the y-axis, and #x=3# rotated around the x axis?

1 Answer
Jim H
Jun 21, 2015

Use disks/washers.

Explanation:

Sketch the region. Note that #2x^2+5# is above (greater than) #x+3#, so the parabola is farther from the axis of rotation.

Therefore:
At a particular #x#, the large radius is, # R = 2x^2+5#, and the small radius is #r = x+3#. The thickness of the disks is #dx#

The volume of each representative disk would be #pi * "radius"^2 * "thickness"#. So the large disk has volume: #pi(2x^2+5)^2 dx# and the small one has volume # pi (x+3)^2 dx#

The volume of the washer is the difference, or #piR^2dx-pir^2dx#and the resulting solid has volume:

#V = pi int_0^3 ((2x^2+5)^2 - (x+3)^2) dx#

# = pi int_0^3 (4x^4+19x^2-6x+16) dx#

You can finish the integral to get # (1932 pi)/5#

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