Method of Cylindrical Shells (Part 3)

In summary, the Method of Cylindrical Shells is an approach for finding the volume of a solid using the equation V = 2π∫<sub>a</sub><sup>b</sup> r(x)h(x)dx, where r(x) is the radius of the shell and h(x) is the height of the shell at a given point x. This method differs from the Method of Disks and Washers in that it involves integrating along the height of the solid instead of the radius. It can be applied to solids with varying cross-sectional shapes, as long as they are circular and the axis of rotation is perpendicular to the axis of the cylinder. The radius of the shell is significant in determining the volume, as
  • #1
shamieh
539
0
The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method..
\(\displaystyle y = -x^2 + 6x - 8, y = 0\)

so I got -8 to 0 for the integral by plotting the graph... How are they getting 2 to 4? You can't solve that by factoring? And when i plugged it into the quadratic formula i got 1 to 5

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shamieh said:
The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method..
\(\displaystyle y = -x^2 + 6x - 8, y = 0\)

so I got -8 to 0 for the integral by plotting the graph... How are they getting 2 to 4? You can't solve that by factoring? And when i plugged it into the quadratic formula i got 1 to 5

nevermind my god I am stupid I forgot to distribute the x

- - - Updated - - -

Actually I take that back, I still don't understand how they get 2 to 4.
 
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  • #2
\(\displaystyle y=-x^2+6x-8=-\left(x^2-6x+8 \right)=-(x-2)(x-4)\)

What is the axis of rotation?
 
  • #3
Oh i was the - sign... I see(Worried)
 
  • #4
MarkFL said:
\(\displaystyle y=-x^2+6x-8=-\left(x^2-6x+8 \right)=-(x-2)(x-4)\)

What is the axis of rotation?

x axis?
 
  • #5
shamieh said:
x axis?

Which method do you feel will be more straightforward, and can you give an element of the volume, that is, the volume of an arbitrary shell, disk or washer (depending on the method you choose)?

I actually recommend doing these problems more than one way when possible just for the practice and as a means of checking your result.
 
  • #6
We can work this type of problem in general by requiring one of the roots of the parabolic boundary to be the origin, and the other on the positive $x$-axis.

So let the region be bounded by:

\(\displaystyle y=-kx(x-a)\) and \(\displaystyle y=0\), which is also the axis of rotation.

Note: \(\displaystyle 0<k\) and \(\displaystyle 0<a\).

Disk method:

The volume of an arbitrary disk is:

\(\displaystyle dV=\pi r^2\,dx\)

where:

\(\displaystyle r=y=-kx(x-a)\)

hence:

\(\displaystyle dV=\pi \left(-kx(x-a) \right)^2\,dx=\pi k^2\left(x^4-2ax^3+a^2x^2 \right)\,dx\)

Summing up the disks, we find:

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

Applying the FTOC, we obtain:

\(\displaystyle V=\pi k^2\left[\frac{1}{5}x^5-\frac{a}{2}x^4+\frac{a^2}{3}x^3 \right]_0^a=\pi a^5k^2\left(\frac{1}{5}-\frac{1}{2}+\frac{1}{3} \right)=\frac{\pi a^5k^2}{30}\)

Shell method:

The volume of an arbitrary shell is:

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

where:

\(\displaystyle r=y\)

To find $h$, consider:

\(\displaystyle y=-kx(x-a)=-kx^2+akx\)

Arranged in standard quadratic form:

\(\displaystyle kx^2-akx+y=0\)

Application of the quadratic formula yields:

\(\displaystyle x=\frac{ak\pm\sqrt{a^2k^2-4ky}}{2k}\)

Hence:

\(\displaystyle h=\frac{ak+\sqrt{a^2k^2-4ky}}{2k}-\frac{ak-\sqrt{a^2k^2-4ky}}{2k}=\frac{\sqrt{a^2k^2-4ky}}{k}\)

And so we have:

\(\displaystyle dV=\frac{2\pi}{k}\left(y\sqrt{a^2k^2-4ky} \right)\,dy\)

To determine the upper limit of integration, we note that the axis of symmetry for the parabola is on \(\displaystyle x=\frac{a}{2}\) and so the upper limit is:

\(\displaystyle y=-k\frac{a}{2}\left(\frac{a}{2}-a \right)=\frac{a^2k}{4}\)

And so the volume is given by:

\(\displaystyle V=\frac{2\pi}{k}\int_0^{\frac{a^2k}{4}} y\sqrt{a^2k^2-4ky}\,dy\)

At this point, we may want to develop a formula for the indefinite integral:

\(\displaystyle I=\int x\sqrt{a+bx}\,dx\)

Let's try the substitution:

\(\displaystyle u=a+bx\,\therefore\,du=b\,dx\)

Hence, the integral becomes:

\(\displaystyle I=\frac{1}{b^2}\int u^{\frac{3}{2}}-au^{\frac{1}{2}}\,du\)

Using the power rule, we find:

\(\displaystyle I=\frac{1}{b^2}\left(\frac{2}{5}u^{\frac{5}{2}}-\frac{2a}{3}u^{\frac{3}{2}} \right)+C=\frac{2}{15b^2}u^{\frac{3}{2}}\left(3u-5a \right)+C\)

Back-substituting for $u$, we obtain:

\(\displaystyle I=\frac{2}{15b^2}(a+bx)^{\frac{3}{2}}\left(3(a+bx)-5a \right)+C\)

\(\displaystyle I=\frac{2}{15b^2}(a+bx)^{\frac{3}{2}}\left(3bx-2a \right)+C\)

Now, applying this to our volume, we get through the application of the FTOC:

\(\displaystyle V=\frac{\pi}{60k^3}\left[\left(a^2k^2-4ky \right)^{\frac{3}{2}}(-12ky-2a^2k^2) \right]_0^{\frac{a^2k}{4}}=\frac{\pi}{60k^3}\left(0+2a^5k^5 \right)\)

And thus:

\(\displaystyle V=\frac{\pi a^5k^2}{30}\)

Applying this formula to the given problem, we then find:

\(\displaystyle V=\frac{\pi 2^51^2}{30}=\frac{16\pi}{15}\)
 

FAQ: Method of Cylindrical Shells (Part 3)

What is the equation for finding the volume of a solid using the Method of Cylindrical Shells?

The equation is V = 2π∫ab r(x)h(x)dx, where r(x) is the radius of the shell and h(x) is the height of the shell at a given point x.

How does the Method of Cylindrical Shells differ from the Method of Disks and Washers?

The Method of Cylindrical Shells involves integrating along the height of the solid, while the Method of Disks and Washers involves integrating along the radius of the solid. This results in different equations and approaches for each method.

Can the Method of Cylindrical Shells be applied to solids with varying cross-sectional shapes?

Yes, the Method of Cylindrical Shells can be applied to solids with varying cross-sectional shapes, as long as the cross-sections are circular and the axis of rotation is perpendicular to the axis of the cylinder.

What is the significance of the radius of the shell in the Method of Cylindrical Shells?

The radius of the shell represents the distance between the axis of rotation and the point on the solid being integrated. It is a crucial factor in determining the volume of the solid using this method.

Are there any limitations to using the Method of Cylindrical Shells?

The Method of Cylindrical Shells can only be applied to solids with circular cross-sections and a perpendicular axis of rotation. It also requires knowledge of the function for the radius and height of the shell, which may not always be easily determined.

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