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Q49E

Expert-verifiedFound in: Page 380

Book edition
2nd

Author(s)
James Stewart

Pages
830 pages

ISBN
9781133112280

**(a) To determine the Cavalieri's Principle.**

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

(a) The Cavalieri's Principle is proved.

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

(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 Summation of moment about any point is Equal to zero.**

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

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**.**

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 \(\underline {\pi {r^2}h} \).

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