# How do you find the volume of the solid generated when the regions bounded by the graphs of the given equations #y=e^-x#, x= -1, x = 2 and the x-axis are rotated about the x-axis?

# VOR = pi/2 (e^2-e^(-4)) #

# " " ~~11.5779320... #

I recommend that you always draw a sketch to clarify what needs calculating.

graph{e^(-x) [-2, 3, -5, 5]}

So in this case:

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To find the volume of the solid generated by rotating the region bounded by the graphs of the equations (y = e^{-x}), (x = -1), (x = 2), and the x-axis about the x-axis, you use the method of cylindrical shells. The volume can be calculated using the following integral:

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

Where (V) represents the volume of the solid, (x) ranges from (-1) to (2), and (2\pi x) represents the circumference of the cylindrical shell at each (x)-value. (e^{-x}) represents the height of each shell. Integrating this expression will give the volume of the solid.

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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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