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djyliett [7]
3 years ago
10

A) Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given cur

ves about the x-axis.
xy = 2, x = 0, y = 2, y = 4
b) Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis.
y = x3, y = 8, x = 0; about x = 9
Mathematics
1 answer:
Leno4ka [110]3 years ago
3 0

(a) See the attached sketch. Each shell will have a radius <em>y</em> chosen from the interval [2, 4], a height of <em>x</em> = 2/<em>y</em>, and thickness ∆<em>y</em>. For infinitely many shells, we have ∆<em>y</em> converging to 0, and each super-thin shell contributes an infinitesimal volume of

2<em>π</em> (radius)² (height) = 4<em>πy</em>

Then the volume of the solid is obtained by integrating over [2, 4]:

\displaystyle 4\pi \int_2^4 y\,\mathrm dy = 2\pi y^2\bigg|_{y=2}^{y=4} = 2\pi (4^2-2^2) = \boxed{24\pi}

(b) See the other attached sketch. (The text is a bit cluttered, but hopefully you'll understand what is drawn.) Each shell has a radius 9 - <em>x</em> (this is the distance between a given <em>x</em> value in the orange shaded region to the axis of revolution) and a height of 8 - <em>x</em> ³ (and this is the distance between the line <em>y</em> = 8 and the curve <em>y</em> = <em>x</em> ³). Then each shell has a volume of

2<em>π</em> (9 - <em>x</em>)² (8 - <em>x</em> ³) = 2<em>π</em> (648 - 144<em>x</em> + 8<em>x</em> ² - 81<em>x</em> ³ + 18<em>x</em> ⁴ - <em>x</em> ⁵)

so that the overall volume of the solid would be

\displaystyle 2\pi \int_0^2 (648-144x+8x^2-81x^3+18x^4-x^5)\,\mathrm dx = \boxed{\frac{24296\pi}{15}}

I leave the details of integrating to you.

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