Let R be the region enclosed by #f(x) = sinx, g(x) =1-x, and x=0#. What is the volume of the solid produced by revolving R around the x-axis?

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Answer 1 · 2017-08-09T21:14:37

Please see below.

Here is a picture of the region with a slice taken perpendicular to the axis of rotation.

enter image source here

Let #c=# the point of intersection of #sinx# and #1-x#.

The volume of the solid is

#V = pi int_0^c ((1-x)^2-sin^2x) dx#

# = pi int_0^c ((1-x)^2 - 1/2(1-cos(2x)) dx#

# = pi[ -(1-x)^3/3-1/2x +1/2sinxcosx]_0^c#

# = pi(-(1-c)^3/3+c/2+1/2sin(c) cos(c)+1/3)#

If desired, we can rewrite using

#sinc = 1-c# and #cosc = sqrt(2c-c^2)#.

Or we can evaluate using #c ~~ 0.51#