# The area under the curve y=e^-x between x=0 and x=1 is rotated about the x axis find the volume?

Volume is

Let us see the graph of

graph{e^(-x) [-2.083, 2.917, -0.85, 1.65]}

To find the desired volume the shaded portion (shown below, will have to be rotated around

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To find the volume of the solid generated by rotating the curve ( y = e^{-x} ) about the x-axis from ( x = 0 ) to ( x = 1 ), we use the method of cylindrical shells. The formula for the volume ( V ) generated by rotating a curve ( y = f(x) ) about the x-axis from ( x = a ) to ( x = b ) is given by:

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

In this case, ( f(x) = e^{-x} ), ( a = 0 ), and ( b = 1 ).

Substituting these values into the formula, we have:

[ V = \int_{0}^{1} 2\pi x e^{-x} , dx ]

Integrating this expression will give us the volume of the solid of revolution.

[ V = 2\pi \int_{0}^{1} x e^{-x} , dx ]

[ = 2\pi \left[ -xe^{-x} - \int_{0}^{1} -e^{-x} , dx \right] ]

[ = 2\pi \left[ -xe^{-x} - \left( -e^{-x} \bigg|_{0}^{1} \right) \right] ]

[ = 2\pi \left[ -xe^{-x} + (1 - e^{-1}) \right] ]

[ = 2\pi (1 - e^{-1}) - 2\pi \int_{0}^{1} xe^{-x} , dx ]

The remaining integral can be evaluated by parts:

[ u = x, \quad dv = e^{-x} , dx ] [ du = dx, \quad v = -e^{-x} ]

[ = 2\pi (1 - e^{-1}) - 2\pi \left[ -xe^{-x} + \int_{0}^{1} -e^{-x} , dx \right] ]

[ = 2\pi (1 - e^{-1}) - 2\pi \left[ -xe^{-x} - e^{-x} \bigg|_{0}^{1} \right] ]

[ = 2\pi (1 - e^{-1}) - 2\pi \left[ -e^{-1} - (1 - e^{-1}) \right] ]

[ = 2\pi (1 - e^{-1}) - 2\pi (-1) ]

[ = 2\pi - 2\pi e^{-1} + 2\pi ]

[ = 2\pi(1 - e^{-1}) + 2\pi ]

[ = 2\pi(1 - e^{-1} + 1) ]

[ = 2\pi(2 - e^{-1}) ]

[ \approx 2.413 , \text{units}^3 ]

So, the volume of the solid generated by rotating the curve ( y = e^{-x} ) about the x-axis from ( x = 0 ) to ( x = 1 ) is approximately ( 2.413 , \text{units}^3 ).

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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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