Let R be the region enclosed by f(x) = x^2 + 2 and g(x) = (x  2)^2. What is the volume of the solid produced by revolving R around the xaxis and then the yaxis?
Those two curves have only one point of intersection. They do not enclose a region.
graph{(y(x^2+2))(y(x2)^2)=0 [19.57, 26.06, 3.74, 19.06]}
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To find the volume of the solid produced by revolving the region enclosed by (f(x) = x^2 + 2) and (g(x) = (x  2)^2) around the xaxis and then the yaxis, you can use the method of cylindrical shells.

First, find the points of intersection between (f(x)) and (g(x)) by setting them equal to each other and solving for (x). These points determine the limits of integration.

Next, integrate (2\pi x \cdot (f(x)  g(x))) with respect to (x) over the interval determined by the points of intersection to find the volume of each shell.

Finally, integrate the resulting expression over the appropriate interval to find the total volume of the solid formed by revolving the region around the xaxis and then the yaxis.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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