Shell Method Problem - about y axis - # 5

Oct 2012
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\(\displaystyle y = 3x^{4}, y = 0, x = 2\)

about \(\displaystyle x = 4\) so revolves about y axis

Find limit of integration

\(\displaystyle 3x^{4} = 0\)

\(\displaystyle x = 0 \)

\(\displaystyle V = 2\pi \int_{0}^{2} (x)(3x^{4}) dx \)

\(\displaystyle V = 2\pi \int_{0}^{2} 3x^{5}) dx\)

\(\displaystyle V = (2\pi) \dfrac{3x^{6}}{6}\) evaluated at 0 and 2

\(\displaystyle V = (2\pi) \dfrac{x^{6}}{2}\) evaluated at 0 and 2

\(\displaystyle V = (2\pi) [[\dfrac{(2)^{6}}{2}] - [0]] = 32 * 2\pi = 64 \pi\) - wrong on homework ?
 

chiro

MHF Helper
Sep 2012
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1,263
Australia
Hey Jason76.

Since it is about the x = 4 axis you will need to shift the function to get the right answer.

This means instead of looking 3x^4 you also need to look at (3x^4 - 4) since you are shifting the function by four units.

Take a look at this for more information:

Shell integration - Wikipedia, the free encyclopedia
 
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Oct 2012
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If were dealing with y axis then it's [function - (shifted value)]

If were dealing with the x axis, then what's the strategy? The article doesn't explain too well.
 

chiro

MHF Helper
Sep 2012
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You shift the y value instead.

The difference between rotation around x and y is that you have a function of the other variable. That's basically it.

If you have say y = f(x) and need x as a function of y then you find the inverse function.

So instead of using say x - a you use y - a for the shift.
 
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Oct 2012
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So instead of using say x - a you use y - a for the shift.

It's

\(\displaystyle a - x \)

and

\(\displaystyle a - y \)

when dealing with shifting.

calculus-2.jpg

calculus.jpg
 
Last edited:
Oct 2012
1,314
21
USA
Would you use the same strategy (a - x), (a - y), regarding revolving around lines (not the y or x axis), with the disk and washer method?