# What is the Volume of Revolution if the area bounded by the curve #y=x^2-4x# and the #x#-axis is is rotated about the #x#-axis?

# (512 pi) /15 \ unit^3#

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

graph{x^2-4x [-10, 10, -5, 5]}

So in this case:

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To find the volume of revolution when the area bounded by the curve (y = x^2 - 4x) and the x-axis is rotated about the x-axis, you can use the disk method. First, determine the bounds of integration by setting the equation (y = x^2 - 4x) equal to zero and solving for x. The roots will give you the bounds of integration. Then, use the formula for the volume of revolution:

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

where (a) and (b) are the bounds of integration, and (f(x)) is the function that defines the curve. In this case, (f(x) = x^2 - 4x). Calculate the integral and the resulting volume will be the volume of revolution.

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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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