# 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?

Please see below.

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

Let

The volume of the solid is

# = 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#

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To find the volume of the solid produced by revolving the region ( R ) enclosed by ( f(x) = \sin(x) ), ( g(x) = 1 - x ), and ( x = 0 ) around the x-axis, you can use the disk method or washer method.

First, find the points of intersection of ( f(x) ) and ( g(x) ):

[ \sin(x) = 1 - x ]

Solve for ( x ) to find the intersection points.

Next, determine which function is on top in the region ( R ) to set up the integral. This will help decide whether to use the disk method or washer method.

Once you have the appropriate setup, use the following formula to find the volume:

[ V = \pi \int_{a}^{b} [f(x)]^2 , dx ]

or

[ V = \pi \int_{a}^{b} ([f(x)]^2 - [g(x)]^2) , dx ]

Here, ( [f(x)]^2 ) represents the area of a disk, and ( ([f(x)]^2 - [g(x)]^2) ) represents the area of a washer.

Evaluate the integral from the lower bound ( a ) to the upper bound ( b ) to find the volume ( V ) of the solid.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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