# Shell Method Problem - about y axis - # 5

#### Jason76

$$\displaystyle y = 3x^{4}, y = 0, x = 2$$

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
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.

Shell integration - Wikipedia, the free encyclopedia

1 person

#### Jason76

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
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.

1 person

#### Jason76

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.

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#### Jason76

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?