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(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 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.

$$

V=\pi r^{2} h

$$

Applications of Integration

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{'transcript': "giving suits solids so you have to solace S. One aimed is to relate a yes one and mm is to be there. So this is a being there because sectional sectional area, it's so by definition of volume of a cylindrical shell we know that the volume of solid one would be the antica from mm to be of E. Yes one the eggs and that's a volume is to be dancing girl. You say this is a 1 B1 A two B two of E. Is one of the eight. If a. S. One it's equal to a. Is soon and A one equal to 82. And we want equal to b. two. Then it implies that We have the NC Girl from A. one the one E. S. One the X. To be equal to A. to b. two of a. Is to the X. So they this implies that the volume three VES one is equal to v. Is to so the volume for both. Uh The same then for the second parts, B B. Buy cava leans principle. The volume of the cylinder of the cylinder is the same as dads of it, writes circular cylinder with radios. Um, and hi, it's right H so this implies that the volume is equal to buy out squared each, hence final results."}

Kwame Nkrumah University of Science and Technology

Applications of Integration