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=6x+7# and #y=x^2# rotated about the line #y=49#?

Question

Paisley Johnson · Accepted Answer

Given a choice, I would use washers, but. . .

Here is a (non-interactive) graph using Desmos (desmos.com)
I have included the line #y=49# in red.

Here is the region using the Socratic grapher. It isn't quite accurate, but you can zoom in and out and drag the graph around

graph{(y-x^2)(y-6x-7) (sqrt(x+1))/(sqrt(x+1))<= 0 [-8.29, 20.21, -6, 8.26]}

#y=6x+7# if and only if #x=1/6y-7/6#

#y=x^2# if and only if #x= +-sqrty#

To use shells we take our representative slices horizontally. So the bounds become

For #y=0# to #y=1#, #x# goes from #-sqrty# to #sqrty#.

From #y-1# to #y=49#, #x# goes from #1/6y-7/6# to #sqrty#.

Throughout the problem, the radius of the cylindrical shell will be #49-y#

So we need to evaluate two integrals:

#2pi int_0^1 (49-y)(sqrty-(-sqrty))dy = 4pi int_0^1 (49y^(1/2)-y^(3/2))dy#

# = 1936/15pi#
And

#2pi int_1^49 (49-y)(sqrty-(1/6y-7/6))dy = 25296/5pi#
The volume is the sum of the two integrals.

Washers

#pi int_-1^7 [(49-x^2)^2-(49-(6x+7))^2]dx = 77824/15pi#

Charles Arocha · Answer

undefined To use the shell method to find the volume of the solid generated by revolving the region bounded by $ y = 6x + 7 $ and $ y = x^2 $ about $ y = 49 $, follow these steps:

1. First, find the points of intersection between the curves $ y = 6x + 7 $ and $ y = x^2 $ by setting them equal to each other and solving for $ x $. This gives $ x^2 = 6x + 7 $.

2. Solve the quadratic equation $ x^2 - 6x - 7 = 0 $ to find the x-coordinates of the intersection points.

3. Next, determine the limits of integration by finding the x-values where the curves intersect. These will be your lower and upper bounds for integration.

4. The radius of each shell will be the distance from the axis of rotation ($ y = 49 $) to the curve $ y = 6x + 7 $ or $ y = x^2 $, depending on which shell you're considering.

5. The height of each shell will be the difference in the y-values of the curves $ y = 6x + 7 $ and $ y = x^2 $ at the corresponding x-value.

6. Set up the integral using the shell method formula: 
$$ V = \int_{a}^{b} 2\pi rh \, dx $$
where $ r $ represents the radius of the shell, $ h $ represents the height of the shell, and $ dx $ represents an infinitesimally small width along the x-axis.

7. Integrate from the lower bound ($ a $) to the upper bound ($ b $) using the expressions for $ r $ and $ h $ in terms of $ x $.

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

Remember to perform any necessary simplifications during the integration process.