# How do you find the volume of a solid that is enclosed by #y=2x+2# and #y=x^2+2# revolved about the x-axis?

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To find the volume of the solid generated by revolving the region between the curves (y = 2x + 2) and (y = x^2 + 2) about the x-axis, we can use the method of cylindrical shells.

The volume (V) of the solid generated by revolving the region between the curves (y = f(x)) and (y = g(x)) from (x = a) to (x = b) about the x-axis is given by:

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

In this case, (f(x) = 2x + 2), (g(x) = x^2 + 2), and we want to find the volume from the points where the two curves intersect. To find these points, we set (f(x) = g(x)) and solve for (x):

[2x + 2 = x^2 + 2] [x^2 - 2x = 0] [x(x - 2) = 0]

So, (x = 0) or (x = 2). These are the points of intersection. Therefore, the limits of integration are (a = 0) and (b = 2).

Now, we can set up the integral:

[V = \int_{0}^{2} 2\pi x [(2x + 2) - (x^2 + 2)] , dx]

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

[V = 2\pi \int_{0}^{2} (2x - x^3) , dx]

[V = 2\pi \left[\frac{1}{2}x^2 - \frac{1}{4}x^4\right]_{0}^{2}]

[V = 2\pi \left[\frac{1}{2}(2)^2 - \frac{1}{4}(2)^4 - \left(\frac{1}{2}(0)^2 - \frac{1}{4}(0)^4\right)\right]]

[V = 2\pi \left[\frac{1}{2}(4) - \frac{1}{4}(16)\right]]

[V = 2\pi \left[2 - 4\right]]

[V = 2\pi (-2)]

[V = -4\pi]

So, the volume of the solid generated by revolving the region between the curves (y = 2x + 2) and (y = x^2 + 2) about the x-axis is (-4\pi).

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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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