Question

Find the volume v of the solid obtained by rotating the region bounded by the given curves about the specified line. Y = 8x3, y = 0, x = 1; about x = 2.

Answer 1

The volume v of the solid obtained is 15.08

By using the cylindrical shell method to find the volume v of the solid which can be expressed by the formula;

• V = 2πrh dr

Then, it is paramount to determine the radius and height of the shell.

From the given information;

• The cylindrical shell formed by the rotation about the line x = 2
• Let consider the radius to be 2 - x
• The circumference = 2π(2 - x)
• The height = 8x³

Then, the volume v of the solid can be estimated by using the cylindrical shell method as follows;

$\mathbf{V = \int ^1_0 2 \pi (2-x) (8x^3) \ dx}$

Open brackets;

$\mathbf{V = 2 \pi \int ^1_0 (16x^3 -8x^4) \ dx}$

Taking the integral form;

$\mathbf{V = 2 \pi \Big [ \dfrac{16x^4}{4}-\dfrac{8x^5}{5} \Big]^1_0}$

Solving the terms in the bracket

$\mathbf{V = 2 \pi \Big [ \dfrac{16(1)^4}{4}-\dfrac{8(1)^5}{5} -0\Big]}$

$\mathbf{V = 2 \pi \Big [ 4-\dfrac{8}{5} \Big]}$

$\mathbf{V = 2 \pi \Big [ \dfrac{20-8}{5} \Big]}$

$\mathbf{V = 2 \pi \Big [ \dfrac{12}{5} \Big]}$

$\mathbf{V = \dfrac{24\pi}{5}}$

$\mathbf{V =15.08}$

Therefore, we can conclude that the volume v of the solid obtained is 15.08

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