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?

Answer 1

See below.

It can be seen from the diagram, that if we form rectangles with a width of #deltax# and a height of #f(x)# and revolve these around the x axis through an angle #pi# radians, we will form a series of discs. These discs will have a radius f(x) and thicness #deltax#. The volume of a disc will therfore be:

#pi(f(x))^2*deltax#

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

We need to use the interval #[0 , 2]#

so our integral will be:

#pi*int_(0)^(2)(e^(1-2x))^2 dx#

#(e^(1-2x))^2=e^(2-4x)#

#pi*int_(0)^(2)(e^(2-4x)) dx=pi*[-1/4e^(2-4x)]_(0)^(2)#

#pi*{[-1/4e^(2-4x)]^(2)-[-1/4e^(2-4x)]_(0)}#

Plugging in upper and lower bounds:

#pi*{[-1/4e^(2-4(2))]^(2)-[-1/4e^(2-4(0))]_(0)}#

#pi*{[-1/(4e^6) ]^(2)-[-1/4e^(2)]_(0)}#

#pi*{(-1+e^8)/(4e^6)}=((-1+e^8)/(4e^6))*picolor(white)(88)# cubic units

Volume #= ((-1+e^8)/(4e^6))*pi~~5.8014#

Volume of Revolution:

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Answer 2

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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Answer from HIX Tutor

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.

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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