Method of Cylindrical Shells Question #2

In summary, the conversation involved using the method of cylindrical shells to find the volume of a solid obtained by rotating the region bounded by the curves y = x^3, y = 8, and x = 0 about the x-axis. The question asked why the y-value was cubed and put in terms of x, and if it was necessary to do so. The answer was that if using the shell method, the volume of an arbitrary shell is 2πrh dy, and since r = y and h = x = y^(1/3), the volume would be 2πy^(4/3) dy. To check the work using the washer method, the volume of an arbitrary washer is π(R^2
  • #1
shamieh
539
0
Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x axis...\(\displaystyle y = x^3\) , \(\displaystyle y = 8\) and \(\displaystyle x = 0\)

So my question is: Why did they cube root the y (to be more technical why did they put it in terms of x? I don't understand what this is accomplishing? Can't you just set up your graph and have a horizontal asymptote at y = 8, a parabola that doesn't pass (2,8), and then just set up your integral and solve as \(\displaystyle 2\pi \int^8_1 x(x^2)\) dx ?

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shamieh said:
Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x axis...\(\displaystyle y = x^3\) , \(\displaystyle y = 8\) and \(\displaystyle x = 0\)

So my question is: Why did they cube root the y (to be more technical why did they put it in terms of x? I don't understand what this is accomplishing? Can't you just set up your graph and have a horizontal asymptote at y = 8, a parabola that doesn't pass (2,8), and then just set up your integral and solve as \(\displaystyle 2\pi \int^8_1 x(x^2)\) dx ?

Nevermind i see what is going on... 2pi * X

so that's why you set x =
 
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  • #2
If you are to use the shell method, you want to observe that the volume of an arbitrary shell is:

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

where:

\(\displaystyle r=y\)

\(\displaystyle h=x=y^{\frac{1}{3}}\)

hence:

\(\displaystyle dV=2\pi y^{\frac{4}{3}}\,dy\)

Summing up all the shells, we find:

\(\displaystyle V=2\pi\int_0^8 y^{\frac{4}{3}}\,dy\)

If you wish to check your work by using the washer method, the volume of an arbitray washer is:

\(\displaystyle dV=\pi\left(R^2-r^2 \right)\,dx\)

where:

\(\displaystyle R=8\)

\(\displaystyle r=y=x^3\)

hence:

\(\displaystyle V=\pi\int_0^2 8^2-x^6\,dx\)

You should verify that both definite integrals give the same result.
 

FAQ: Method of Cylindrical Shells Question #2

What is the method of cylindrical shells?

The method of cylindrical shells is a technique used in integral calculus to calculate the volume of a solid of revolution. It involves dividing the solid into thin cylindrical shells and using the formula V = 2πrh to find the volume of each shell, then integrating the volumes to find the total volume of the solid.

When is the method of cylindrical shells used?

The method of cylindrical shells is typically used when the solid being revolved has a cross-sectional area that can be expressed as a function of the height or radius of the solid. This method is particularly useful for finding the volume of objects such as cylinders, cones, and spheres.

How is the method of cylindrical shells different from the method of disks/washers?

The method of cylindrical shells and the method of disks/washers are both used to find the volume of solids of revolution, but they differ in the shape of the slices used to approximate the solid. The method of cylindrical shells uses cylindrical slices, while the method of disks/washers uses circular slices.

What are the limitations of the method of cylindrical shells?

The method of cylindrical shells can only be used to find the volume of solids of revolution with a known axis of rotation. It also requires the solid to have a consistent cross-sectional area that can be expressed as a function of the height or radius of the solid.

Can the method of cylindrical shells be used for non-circular solids?

Yes, the method of cylindrical shells can be used for non-circular solids of revolution as long as the cross-sectional area can be expressed as a function of the height or radius of the solid. This includes shapes such as ellipsoids, paraboloids, and hyperboloids.

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