Question

Find the volume v of the solid obtained by rotating the region bounded by the given curves about the specified line. Y = 27x3, y = 0, x = 1; about x = 2.

Answer 1

The volume of the solid of revolution is $\frac{1674\pi}{5}$ cubic units.

The volume of the solid of revolution can be determined by the following integral equation:

$V = 2\pi\int\limits^a_b {|f(x)-g(x)|} \, dx$ (1)

Where:

• $g(x)$ - Lower bound function
• $f(x)$ - Upper bound function
• $a$ - Upper bound
• $b$ - Lower bound

If we know that $b = 2$, $b = 1$, $g(x) = 0$ and $f(x) = 27\cdot x^{3}$, then the volume of the solid of revolution is:

$V = 2\pi\int\limits^2_1 {x\cdot |27\cdot x^{3}-0|} \, dx$

$V = 54\pi \int\limits^2_1 {x^{4}} \, dx$

$V = 54\pi \left(\frac{2^{5}}{5}-\frac{1^{5}}{5} \right)$

$V = \frac{1674\pi}{5}$

The volume of the solid of revolution is $\frac{1674\pi}{5}$ cubic units.

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