# How do you find the volume of the solid obtained by rotating the region bounded by the curves #y=x^2# and #y=2-x^2# and #x=0# about the line #x=1#?

I would use shells (to avoid doing two integrals).

Here is a picture of the region. I have included the axis of rotation (

When we rotate the slice, we'll get a cylindrical shell.

The representative cylindrical shell will have volume:

#2 pixx"radius"xx"height"xx"thickness"# .Here we have:

#"radius"=r=(1-x)# .The height will be the greater

#y# value minus the lesser (the top y minus the bottom y).

#"height" = y_"top"-y_"bottom" = (2-x^2)-(x^2) = (2-2x^2)#

#"thickness" = dx# We also not that in the region,

#x# varies from#0# to#1# Using shell, the volume of the solid is found by evaluating:

#V - int_a^b 2 pixx"radius"xx"height"xx"thickness"# So we need

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

# = 2piint_0^1 (2-2x-2x^2+2x^3) dx#

# = 5/3 pi#

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To find the volume of the solid obtained by rotating the region bounded by the curves (y=x^2) and (y=2-x^2) about the line (x=1), you can use the method of cylindrical shells or the method of washers. Both methods involve integrating the cross-sectional area along the axis of rotation.

Using the method of cylindrical shells, you would integrate (2\pi x \cdot h(x) dx), where (h(x)) represents the height of the shell at each point along the axis of rotation.

Using the method of washers, you would integrate (\pi [R(x)^2 - r(x)^2] dx), where (R(x)) is the outer radius and (r(x)) is the inner radius of each washer at a given x-coordinate.

First, you would determine the limits of integration by finding the x-values at which the curves intersect. Then, you would calculate the outer and inner radii at each x-value. Once you have the expressions for the outer and inner radii, you can proceed with the chosen method of integration to find the volume of the resulting 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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