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#?
Given a choice, I would use washers, but. . .
Here is a (noninteractive) graph using Desmos (desmos.com)
I have included the line
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{(yx^2)(y6x7) (sqrt(x+1))/(sqrt(x+1))<= 0 [8.29, 20.21, 6, 8.26]}
To use shells we take our representative slices horizontally. So the bounds become
For
From
Throughout the problem, the radius of the cylindrical shell will be
So we need to evaluate two integrals:
# = 1936/15pi# And
#2pi int_1^49 (49y)(sqrty(1/6y7/6))dy = 25296/5pi# The volume is the sum of the two integrals.
Washers
#pi int_1^7 [(49x^2)^2(49(6x+7))^2]dx = 77824/15pi#
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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:

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 ).

Solve the quadratic equation ( x^2  6x  7 = 0 ) to find the xcoordinates of the intersection points.

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

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.

The height of each shell will be the difference in the yvalues of the curves ( y = 6x + 7 ) and ( y = x^2 ) at the corresponding xvalue.

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 xaxis.

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

Evaluate the integral to find the volume of the solid.
Remember to perform any necessary simplifications during the integration process.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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