Questions from General Calculus

Q: (a) Cavalieri’s Principle states that if a family of parallel

(a) Cavalieri’s Principle states that if a family of parallel planes gives equal cross-sectional areas for two solidsS1 and S2, then the volumes of S1 and S2 are equal. Prove this pr...

Q: Find the volume common to two circular cylinders, each with radius

Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles.

Q: Find the volume common to two spheres, each with radius r

Find the volume common to two spheres, each with radius r, if the center of each sphere lies on the surface of the other sphere.

Q: A bowl is shaped like a hemisphere with diameter 30 cm.

A bowl is shaped like a hemisphere with diameter 30 cm. A heavy ball with diameter 10 cm is placed in the bowl and water is poured into the bowl to a depth of h centimeters. Find the volume of water i...

Q: A hole of radius r is bored through the middle of a

A hole of radius r is bored through the middle of a cylinder of radius R < r at right angles to the axis of the cylinder. Set up, but do not evaluate, an integral for the volume cut out.

Q: A hole of radius r is bored through the center of a

A hole of radius r is bored through the center of a sphere of radius R > r. Find the volume of the remaining portion of the sphere.

Q: Sketch the region enclosed by the given curves. Decide whether to

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the reg...

Q: Some of the pioneers of calculus, such as Kepler and Newton

Some of the pioneers of calculus, such as Kepler and Newton, were inspired by the problem of finding the volumes of wine barrels. (In fact Kepler published a book Stereometria doliorum in 1615 devoted...

Q: Suppose that a region / has area A and lies above the

Suppose that a region / has area A and lies above the x-axis. When / is rotated about the x-axis, it sweeps out a solid with volume V1. When / is rotated about the line y = -k (where k is a positive n...

Q: Find the volume of the solid obtained by rotating the region bounded

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.