Question

Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by y=x^8 and y = 256 in the first quadrant about the y-axis.

Answer 1

Using the shell method, the volume integral would be

$\displaystyle 2\pi \int_0^2 x(256-x^8)\,\mathrm dx$

That is, each shell has a radius of x (the distance from a given x in the interval [0, 2] to the axis of revolution, x = 0) and a height equal to the difference between the boundary curves y = x ⁸ and y = 256. Each shell contributes an infinitesimal volume of 2π (radius) (height) (thickness), so the total volume of the overall solid would be obtained by integrating over [0, 2].

The volume itself would be

$\displaystyle 2\pi \int_0^2 x(256-x^8)\,\mathrm dx = 2\pi \left(128x^2-\frac1{10}x^{10}\right)\bigg|_{x=0}^{x=2} = \boxed{\frac{4096\pi}5}$

Using the disk method, the integral for volume would be

$\displaystyle \pi \int_0^{256} \left(\sqrt[8]{y}\right)^2\,\mathrm dy = \pi \int_0^{256} \sqrt[4]{y}\,\mathrm dy$

where each disk would have a radius of x = ⁸√y (which comes from solving y = x ⁸ for x) and an infinitesimal height, such that each disk contributes an infinitesimal volume of π (radius)² (height). You would end up with the same volume, 4096π/5.

Answer 2

The volume of the solid formed by revolving the region bounded by y=x^8 and y = 256 in the first quadrant about the y-axis is 4096π/5 cubic units.

What is integration?

It is defined as the mathematical calculation by which we can sum up all the smaller parts into a unit.

We have a function:

$\rm y = x^8$   or

$x = \sqrt[8]{y}$

And  y = 256

By using the vertical axis of rotation method to evaluate the volume of the solid formed by revolving the region bounded by the curves.

$\rm V = \pi \int\limits^a_b {x^2} \, dy$

Here a = 256, b = 0, and $x = \sqrt[8]{y}$

$\rm V = \pi \int\limits^{256}_0 {(\sqrt[8]{y}^2) } \, dy$

After solving definite integration, we will get:

$\rm V = \pi(\frac{4096}{5} )$  or

$\rm V =\frac{4096}{5}\pi$ cubic unit

Thus, the volume of the solid formed by revolving the region bounded by y=x^8 and y = 256 in the first quadrant about the y-axis is 4096π/5 cubic units.

Learn more about integration here:

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