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 = e^ (x)#, bounded by: #y = 0#, #x = 1#, #x = 0# rotated about the #x=1#?
This looks like:
graph{(y  e^(x))(y)(x + 1)(sqrt(0.25  (x + 0.5)^2))/(sqrt(0.25  (x + 0.5)^2)) <= 0 [3.29, 5.48, 0.855, 3.52]}
Therefore, you have:
Overall we have:
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To 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 = e^{x} ), bounded by ( y = 0 ), ( x = 1 ), ( x = 0 ) rotated about the line ( x = 1 ), follow these steps:

Determine the axis of rotation. In this case, it's ( x = 1 ).

Draw a sketch of the region and the axis of rotation to visualize the problem.

Identify the bounds of integration. In this case, ( x ) goes from (1) to (0).

Choose a representative horizontal rectangle parallel to the axis of rotation. Each rectangle has height ( e^{x} ) and width ( dx ).

The radius of the shell is the distance from the rectangle's side to the axis of rotation, which is ( 1  x ).

The volume element of each shell is ( 2\pi (1  x) e^{x} , dx ).

Integrate the volume element over the given bounds of integration:
[ V = \int_{1}^{0} 2\pi (1  x) e^{x} , dx ]
 Evaluate the integral to find the volume of the solid generated by revolving the given region about ( x = 1 ).
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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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