How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region #y=8-x^2# #y=x^2# #x=0# rotated about the y axis?

Question

Christopher Allers · Accepted Answer

#16pi#

The volume of the solid generated would be calculated in two parts. The two functions are parabolas one opening up and the other opening down on the same axis of symmetry x=0. The figure is shown alongside. The point of intersection is (2,4) on the right side and (-2,4) on the left side. The elementary shells would be formed along the line y=4. The volume integrals would be

#int_0^2 2pix dx (4-x^2) + int_0^2 2pix dx (8-x^2-4)#
(volume of a shell on green) (volume of shell on red parabola)
parabola)

=#4pi int_0^2 (4x-x^3)dx#

=#4pi[2x^2 -x^4 /4]_0^2#

= #16pi#

#int_0^2 2pix dx (4-x^2) + int_0^2 2pix dx (8-x^2-4)#
(volume of a shell on green) (volume of shell on red parabola)
parabola)

=#4pi int_0^2 (4x-x^3)dx#

=#4pi[2x^2 -x^4 /4]_0^2#

= #16pi#

Paisley Johnson · Answer

undefined To use the shell method to set up and evaluate the integral for finding the volume of the solid generated by revolving the given plane region about the y-axis, follow these steps:

1. Visualize the solid generated by revolving the region $ y = 8 - x^2 $, $ y = x^2 $, and $ x = 0 $ about the y-axis.

2. Determine the limits of integration by finding the intersection points of the curves $ y = 8 - x^2 $ and $ y = x^2 $. These points will give you the range of y-values over which you need to integrate.

3. Identify a representative shell. Since we are revolving the region about the y-axis, the representative shell will be vertical. Its height will be $ \Delta y $, and its radius will be the distance from the y-axis to the curve.

4. Express the volume of the representative shell as $ dV = 2\pi r h \, dy $, where $ r $ is the radius of the shell and $ h $ is its height.

5. Express $ r $ and $ h $ in terms of $ y $ using the given equations of the curves.

6. Integrate $ dV $ from the lower limit of integration to the upper limit of integration to find the total volume:

$$
V = \int_{y_{\text{lower}}}^{y_{\text{upper}}} 2\pi r h \, dy
$$

7. Evaluate the integral to find the volume of the solid.

This process will give you the integral that represents the volume of the solid generated by revolving the given region about the y-axis using the shell method.