Sam m's question at Yahoo Answers regarding a solid of revolution

In summary, the volume of the solid generated by revolving around the line x = 4, given the region bounded by y=x^2+2, y=1, x=0, x=1 is approximately 9.17 cubic units.
  • #1
MarkFL
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MHB
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Here is the question:

Find the volume of the solid generated by revolving around the line x = 4?

given the region bounded by y=x^2+2, y=1, x=0, x=1?

I have posted a link there to this thread so the OP can view my work.
 
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  • #2
Hello sam m,

Let's try the shell method first. The volume of an arbitrary shell is:

\(\displaystyle dV=2\pi rh\,dx\)

where:

\(\displaystyle r=4-x\)

\(\displaystyle h=x^2+2-1=x^2+1\)

And so we have:

\(\displaystyle dV=2\pi(4-x)\left(x^2+1 \right)\,dx=2\pi\left(-x^3+4x^2-x+4 \right)\,dx\)

Hence, summing the shells, we find:

\(\displaystyle V=2\pi\int_0^1 -x^3+4x^2-x+4\,dx\)

Application of the FTOC yields:

\(\displaystyle V=2\pi\left[-\frac{1}{4}x^4+\frac{4}{3}x^3-\frac{1}{2}x^2+4x \right]_0^1=2\pi\left(-\frac{1}{4}+\frac{4}{3}-\frac{1}{2}+4 \right)=\frac{55}{6}\pi\)

Now, we can check our answer by using the washer method. And we can ease our calculations by observing that for $1\le y\le2$ the inner and outer radii are constant, so we have a washer whose thickness is 1 unit and its ourter radius is 4and its inner radius is 3, so this part of the volume is:

\(\displaystyle V_1=\pi\left(4^2-3^2 \right)(1)=7\pi\)

Now, for $2\le y\le3$ we find the outer radius varies, and so the volume of an arbitrary washer is:

\(\displaystyle dV_2=\pi\left(R^2-r^2 \right)\,dy\)

where:

\(\displaystyle R=4-x=4-\sqrt{y-2}\)

\(\displaystyle r=3\)

And so we have:

\(\displaystyle dV_2=\pi\left(\left(4-\sqrt{y-2} \right)^2-3^2 \right)\,dy=\pi\left(y-8\sqrt{y-2}+5 \right)\,dy\)

Adding the washers, we obtain:

\(\displaystyle V_2=\pi\int_2^3 y-8\sqrt{y-2}+5\,dy\)

Application of the FTOC yields:

\(\displaystyle V_2=\pi\left[\frac{1}{2}y^2-\frac{16}{3}(y-2)^{\frac{3}{2}}+5y \right]_2^3=\pi\left(\left(\frac{9}{2}-\frac{16}{3}+15 \right)-\left(2+10 \right) \right)=\pi\left(\frac{85}{6}-12 \right)=\frac{13}{6}\pi\)

Adding the two volumes to get the total:

\(\displaystyle V=V_1+V_2=7\pi+\frac{13}{6}\pi=\frac{55}{6}\pi\)

And so, this checks with the shell method. Thus we find the volume of the solid of revolution in units cubed is:

\(\displaystyle V=\frac{55}{6}\pi\)
 

FAQ: Sam m's question at Yahoo Answers regarding a solid of revolution

1. What is a solid of revolution?

A solid of revolution is a three-dimensional object that is formed by rotating a two-dimensional shape around an axis.

2. What is the purpose of finding the volume of a solid of revolution?

The volume of a solid of revolution is important in various fields such as engineering, physics, and mathematics. It allows us to calculate the amount of space occupied by the object and can be used to solve real-world problems.

3. How do you find the volume of a solid of revolution?

The volume of a solid of revolution can be found by using the disk method or the shell method. The disk method involves slicing the object into thin disks and adding up their volumes, while the shell method involves slicing the object into hollow cylinders and adding up their volumes.

4. What shapes can be used to form a solid of revolution?

Any two-dimensional shape can be used to form a solid of revolution, as long as it is rotated around an axis. Some common shapes used are circles, squares, and triangles.

5. Are there any real-life applications of solids of revolution?

Yes, solids of revolution have various real-life applications such as in the design of bottles, pipes, and roller coasters. They are also used in calculating the volume of objects in manufacturing and construction industries.

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