How do you find the volume of the solid generated by revolving the region bounded by the graphs #y=e^(x/2), y=0, x=0, x=4#, about the x axis?

Answer 1

Please see below.

Here is a graph of the region in blue. A slice has been taken perpendicular to the axis of rotation. The rotation as shown by the arrow/arc.

The representative slice is a disc of

thickness dx

and radius

#r =y_"greater" - y_"lesser" = e^(x/2)-0 = e^(x/2)# .

The volume of the representative slice (disc) is

#pir^2"thickness" = pi (e^(x/2))^2 dx = pi e^x dx# .

The values of #x# vary from #0# to #4#, so the resulting solid has volume

#V = int_0^4 pi e^x dx#.

Evaluate the integral to get

#V = pi(e^4-1) ~~ 53.6# (cubic units).

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Answer 2

To find the volume of the solid generated by revolving the region bounded by the graphs ( y = e^{x/2} ), ( y = 0 ), ( x = 0 ), and ( x = 4 ) about the x-axis, you can use the method of cylindrical shells.

The formula for the volume of the solid generated by revolving the region bounded by the graphs of two functions ( y = f(x) ) and ( y = g(x) ) (where ( f(x) ) is the upper function and ( g(x) ) is the lower function) about the x-axis from ( x = a ) to ( x = b ) is given by:

[ V = \int_{a}^{b} 2\pi x (f(x) - g(x)) , dx ]

In this case, ( f(x) = e^{x/2} ), ( g(x) = 0 ), ( a = 0 ), and ( b = 4 ).

So, the volume ( V ) is:

[ V = \int_{0}^{4} 2\pi x (e^{x/2} - 0) , dx ]

You can then integrate this expression to find the volume of the solid.

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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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