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Method of Cylindrical Shells Question #2

shamieh

Active member
Sep 13, 2013
539
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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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 thats why you set x =
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
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.