How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y=5e^(x)# and #y=5e^(x)#, x = 1, about the y axis?
Slicing to cylindricalshell elements for integration gives approximation only. Circularannular elements are used. To be continued, in the 2nd answer.
See graph to see the area that revolves about yaxis ( x = 0 ). graph{ (y5(2.718)^x)(y  5(2.718)^(x))(x1+0y)=0[0 1.1 0 13.6]} The curves meet at A(5, 0).
They meet x = 1 at B( 1, 5 / e ) and C(1, 5e ).
Inversely, the equations are
setting limits for integration with respect to y.
The area ls divided into two parts;
y from 5 / e to 5
Likewise,
with y from 5 to 5 e.
Note that the integrand is the same function of y, for both.So,
with y from 5/e to 5e. Use integration by parts method.
between 5 / e and 5 e
between the limits
between the limits.
I would review my answer for corrections, if any.
The easier cylindricalshell elements for integration,
applied to right circular cone of height 1 and base radius 1, gives
.
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Continuation, for the 2nd part.
Answer:
between the limits y between 5 / e and 5 e. ( Use ln e = 1. )
At the upper limit, the value is
At the lower limit, this becomes
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To use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by ( y = 5e^x ), ( y = 5e^{x} ), and ( x = 1 ) about the yaxis, follow these steps:

Determine the interval of integration. In this case, it's from ( x = 0 ) to ( x = 1 ) since the intersection point of ( y = 5e^x ) and ( y = 5e^{x} ) is at ( x = 0 ).

Set up the integral for the volume using cylindrical shells formula: [ V = 2\pi \int_{0}^{1} x \cdot h(x) , dx ]

Find the height function, ( h(x) ), which represents the height of each cylindrical shell. In this case, it's the difference between the top curve and the bottom curve: [ h(x) = (5e^x)  (5e^{x}) ]

Now, substitute ( h(x) ) into the integral: [ V = 2\pi \int_{0}^{1} x \cdot ((5e^x)  (5e^{x})) , dx ]

Integrate with respect to ( x ) over the interval ([0, 1]) to find the volume.
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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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