# The region under the curves #y=e^(1-2x), 0<=x<=2# is rotated about the x axis. How do you sketch the region and find the volumes of the two solids of revolution?

See below.

It can be seen from the diagram, that if we form rectangles with a width of

When these discs are summed in the given interval, they will give the volume of revolution.

We need to use the interval

so our integral will be:

Plugging in upper and lower bounds:

Volume

Volume of Revolution:

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To sketch the region, plot the curve (y = e^{1-2x}) for (0 \leq x \leq 2). This curve starts at (x = 0) with (y = e), then decreases as (x) increases, approaching (y = 0) as (x) approaches 2. The region bounded by this curve and the x-axis between (x = 0) and (x = 2) is the area to be rotated.

To find the volumes of the two solids of revolution, consider splitting the region into two parts at (x = 1), where the curve crosses the line (x = 1).

For the region where (0 \leq x \leq 1), you're rotating the curve about the x-axis, resulting in one solid. Use the formula for the volume of a solid of revolution about the x-axis.

For the region where (1 \leq x \leq 2), you're also rotating the curve about the x-axis, resulting in another solid. Again, use the formula for the volume of a solid of revolution about the x-axis.

Integrate the appropriate functions to find the volumes of these solids.

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