How do you find the volume bounded by #y=3-x^2# and y=2 revolved about the y=2?

Question

Isabella Ackerman · Accepted Answer

Volume #=(16pi)/15# cu.units

(Sorry if this is in excessive detail)
The given regions in the Cartesian plane look like:

If this is rotated about #y=2#,
we have:

For ease of calculation, it is convenient if we shift this solid down so the axis of rotation falls along the X-axis:

Note that the radius (relative to the X-axis) of this shifted volume is
equal to #x^2-1# for #x in [-1,+1]#
and we can slice this solid into thin disks, each with a thickness of #delta#,
so that each disk has a volume of #delta * prr^2=pi * delta* (x^2-1)^2#

With very small values of #delta# the sum of the volumes of all such disks will be the volume of the rotated solid.

We can evaluate this sum with #deltararr0# using the integral:
#color(white)("XXX")int_(-1)^(+1) pi * (x^2-1)^2 dx#

#color(white)("XXX")=pi * int_(-1)^(+1) x^4-2x^2+1 dx#

#color(white)("XXX")=pi * ((x^5)/5-(2x^3)/3+x|_(-1)^(+1))#

#color(white)("XXX")=pi* ( (3-10+15)/15-((-3)-(-10)+(-15))/15)#

#color(white)("XXX")=pi * (8/15-(-8)/15)#

#color(white)("XXX")=16/15pi#

Ezra Adams · Answer

undefined To find the volume bounded by the curves $ y = 3 - x^2 $ and $ y = 2 $ revolved about the line $ y = 2 $, you can use the method of cylindrical shells.

1. Determine the limits of integration by finding the points of intersection between the curves $ y = 3 - x^2 $ and $ y = 2 $.
2. Set up the integral for the volume using the formula $ V = \int_{a}^{b} 2\pi x f(x) \, dx $, where $ f(x) $ represents the distance between the curves and $ x $ varies from the leftmost intersection point to the rightmost.
3. Evaluate the integral to find the volume of the solid generated by revolving the region between the curves about the line $ y = 2 $.

Axel Alvarez · Answer

undefined To find the volume bounded by the curves $ y = 3 - x^2 $ and $ y = 2 $ revolved about the line $ y = 2 $, you can use the method of cylindrical shells.

1. Determine the limits of integration by finding the points of intersection between the curves. Set $ 3 - x^2 = 2 $ and solve for $ x $ to find the points of intersection.
2. Set up the integral using the formula for the volume of cylindrical shells: $ V = \int_{a}^{b} 2\pi rh \, dx $, where $ r $ is the distance from the axis of rotation to the shell, and $ h $ is the height of the shell.
3. Express $ r $ and $ h $ in terms of $ x $.
4. Integrate with respect to $ x $ from the lower limit of integration $ a $ to the upper limit of integration $ b $.

This will give you the volume of the solid formed by revolving the region bounded by the curves about the line $ y = 2 $.