(a) Cavalieri's Principle states that if a family of parallel planes gives equal cross-section areas for two solids $ S_1 $ and $ S_2 $ then the volumes of $ S_1 $ and $ S_2 $ are equal. Prove this principle.

(b) Use Cavalieri's Principle to find the volume of the oblique cylinder shown in the figure.

(a) Volume $\left(S_{1}\right)=\int_{0}^{h} A(z) d z=$ Volume $\left(S_{2}\right)$ since the cross-sectional area $A(z)$ at height $z$ is the same for both solids.

(b) By Cavalieri's Principle, the volume of the cylinder in the figure is the same as that of a right circular cylinder with radius $r$ and height $h,$ that is, $\pi r^{2} h$

Applications of Integration

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