Calculus-Sec-6-2-Solution

6.2   Volumes                                          by SGLee - HSKim, YJLim

1-17. Calculate the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

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18-23. Set up, but do not evaluate, an integral for the volume of the solid obtained

by rotating the region bounded by the given curves about the specified line.

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24-27. Describe the solid of revolution whose volume is represented by the integral.

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28-32. Find the volume of the described solid .

28. A right circular cone with height h and base radius r.

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29. A frustum of a right circular cone with height h, lower base radius R, and top radius r.

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30. A tetrahedron with three mutually perpendicular faces and

three mutually perpendicular edges with lengths 6cm, 8cm and 10cm.

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31. The base of S is an elliptical region with boundary curve .

Cross-sections perpendicular to the -axis are isosceles right triangles with the hypotenuse in the base.

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32. A bowl is shaped like a hemisphere with diameter 40 cm. A ball with diameter 20 cm is placed in the bowl

and water is poured into the bowl to a depth of  centimeters. Find the volume of water in the bowl.

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34-37. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

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38. Find the volume of the solids obtained by rotating the region bounded by

the curves  and  about the following lines:

(a) the -axis.

(b) the -axis.

(c) .

(a)

(b)

(c)

39. Let  be the region in the first quadrant bounded by the curves  and .

Calculate the following quantities.

(a) The area of .

(b) The volume obtained by rotating  about the -axis.

(c) The volume obtained by rotating  about the -axis.

(a)

(b)

(c)

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