# The region is bounded by the given curves #y=x, y=4-x, 0<=x<=2# is rotated about the x-axis, how do you find the volume of the two solids of revolution?

See below.

I am reading this as the volume of the shaded area A rotated around the x axis, and the volume of shaded area B rotated around the x axis. First we find the volume of B, then we find the volume of A+B and subtract volume B from this to find the volume A

Volume of B

Volume of A + B

Volume of A = (A + B) - B.

So:

Volume of A =

Volume of B =

Revolution of A:

Revolution of B:

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To find the volume of the two solids of revolution, you first need to find the points of intersection of the curves y = x and y = 4 - x. These points are (2, 2) and (0, 4).

For the first solid, the outer curve is y = 4 - x and the inner curve is y = x.

For the second solid, the outer curve is y = x and the inner curve is y = 4 - x.

Using the disk method, the volume of each solid can be calculated by integrating the area of the cross-sections perpendicular to the x-axis.

For the first solid: [V_1 = \pi \int_0^2 (4-x)^2 - x^2 ,dx]

For the second solid: [V_2 = \pi \int_0^2 x^2 - (4-x)^2 ,dx]

Integrate each expression over the given interval to find the volume of each 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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