# How do you find the volume of the region bounded #y = x²# and #y =1# is revolved about the line# y = -2#?

See the explanation.

Here is the region (in blue) with the line

A representative slice in black and the two radii (vertical red lines).

The volume of a washer is

Where

In this question,

Expand the polynomial integrand and evaluate to get:

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To find the volume of the region bounded by (y = x^2) and (y = 1) when revolved about the line (y = -2), you can use the method of cylindrical shells. The formula for the volume using cylindrical shells is:

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

where (f(x)) is the function defining the outer radius, and (a) and (b) are the bounds of integration.

In this case, (f(x)) is the distance from the axis of rotation ((y = -2)) to the outer curve ((y = 1)), which is (1 - (-2) = 3).

So, (f(x) = 3) for all (x) in the interval ([0, 1]).

Substituting this into the formula:

[ V = 2\pi \int_{0}^{1} x \cdot 3 , dx ]

Now, integrate this expression within the given bounds to find 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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