Question

(a) To determine the Cavalieri's Principle.

(b) To determine the volume of the oblique cylinder using Cavalieri's principle.

Short answer

(a) The Cavalieri's Principle is proved.

(b) The volume of the oblique cylinder by the use of Cavalieri's principle is \(\pi {r^2}h\).

Step-by-step solution

Given

(a) Cavalieri's Principle states that if a family of parallel planes gives equal cross-sectional areas for two solids \({S_1}\) and \({S_2}\), then the volume of \({S_1}\) and \({S_2}\) are equal.

(b) The oblique cylinder of radius \(r\) and height \(h\).

The Concept ofmoment and forces

The Summation of moment about any point is Equal to zero.

The Summation of forces along any direction is equal to zero.

Evaluate the Cavalieri's Principle (a)

The expression to find the volume of the parallel planes as shown below:

\(V = \int_0^h A (z)dz\) …………….. (1)

Refer to Equation (1).

Consider that the volume of solid\({S_1}\)and\({S_2}\)are equal.

\(\begin{aligned}{}V\left( {{S_1}} \right) = V\left( {{S_2}} \right)\\ = \int_0^h A (z)dz\end{aligned}\)

Hence, the Cavalieri's Principle is proved.

Evaluate the volume of the oblique cylinder(b)

The area of the oblique cylinder as shown below.

\(A(z) = \pi {r^2}\)

Find the volume of the oblique cylinder by the use of Cavalieri's principle as shown below.

Substitute \(\pi {r^2}\) for \(A(z)\) in Equation (1).

\(\begin{aligned}{}V = \int_0^k \pi {r^2}dz\\ = \pi {r^2}(z)_0^h\\ = \pi {r^2}(h - 0)\\ = \pi {r^2}h\end{aligned}\)

Therefore, the volume of the oblique cylinder by the use of Cavalieri's principle is \(\underline {\pi {r^2}h} \).