What is the volume of the solid?

Question

The region in quadrant 1 under the graph of #y=4-x^2# is rotated about the line to form a solid. Find the volume of the solid.

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Answer 1 · 2018-05-22T16:28:45

Rotating about the #x-#axis: #V = 53.62 " units"^3#

Given: #y = 4-x^2# rotate about a line

Since a line isn't specified, rotate about the #x#-axis:

graph{4-x^2 [-.5, 10, -1, 5]}

Using the disk method:

#V = pi int_0^2 (4-x^2)^2 dx#

#V = pi int_0^2 (16 - 8x^2 + x^4) dx#

#V = pi [16x - 8/3 x^3 + 1/5 x^5]_0^2#

#V = pi (32 - 64/3 + 32/5 - 0)#

#V = 256/15 pi ~~ 53.62 " units"^3#