# The region under the curve #y=1/x# bounded by #1<=x<=2# is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?

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To sketch the region bounded by the curve (y = \frac{1}{x}) and the limits (1 \leq x \leq 2), you first plot the graph of (y = \frac{1}{x}) within the specified interval. This curve will be in the first quadrant, starting from a point close to the y-axis and curving upwards as x increases. The region of interest lies between the curve and the x-axis, bound by the vertical lines (x = 1) and (x = 2).

a) To rotate this region about the x-axis, you visualize rotating the region beneath the curve around the x-axis, forming a solid with a hole through the center. This solid is shaped like a right circular cone. To find its volume, you use the method of cylindrical shells, which yields:

[ V = \pi \int_{1}^{2} \left( \frac{1}{x} \right)^2 dx ]

b) To rotate the region about the y-axis, you visualize rotating the region beneath the curve around the y-axis. This forms a solid resembling a horn-like shape. To find its volume, you use the method of disks or washers. The formula is:

[ V = \pi \int_{1}^{2} \left( \frac{1}{x} \right)^2 dx ]

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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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