Questions from General Calculus

Q: (a) Show that the volume of a segment of height

(a) Show that the volume of a segment of height h of a sphere of radius r is (b) Show that if a sphere of radius 1 is sliced by a plane at a distance x from the center in such a way that the volume o...

Q: Archimedes’ Principle states that the buoyant force on an object partially or

Archimedes’ Principle states that the buoyant force on an object partially or fully submerged in a fluid is equal to the weight of the fluid that the object displaces. Thus, for an o...

Q: Water in an open bowl evaporates at a rate proportional to the

Water in an open bowl evaporates at a rate proportional to the area of the surface of the water. (This means that the rate of decrease of the volume is proportional to the area of the surface.) Show t...

Q: A sphere of radius 1 overlaps a smaller sphere of radius r

A sphere of radius 1 overlaps a smaller sphere of radius r in such a way that their intersection is a circle of radius r. (In other words, they intersect in a great circle of the small sphere.) Find r...

Q: The figure shows a curve C with the property that, for

The figure shows a curve C with the property that, for every point P on the middle curve y = -x2, the areas A and B are equal. Find an equation for C.

Q: A paper drinking cup filled with water has the shape of a

A paper drinking cup filled with water has the shape of a cone with height h and semi- vertical angle

Q: A clepsydra, or water clock, is a glass container with

A clepsydra, or water clock, is a glass container with a small hole in the bottom through which water can flow. The “clock” is calibrated for measuring time by plac...

Q: A cylindrical container of radius r and height L is partially filled

A cylindrical container of radius r and height L is partially filled with a liquid whose volume is V. If the container is rotated about its axis of symmetry with constant angular speed

Q: Suppose the graph of a cubic polynomial intersects the parabola y =

Suppose the graph of a cubic polynomial intersects the parabola y = x2 when x = 0, x = a, and x = b, where 0 < a < b. If the two regions between the curves have the same area, how is b related to a?

Q: Sketch the region enclosed by the given curves and find its area

Sketch the region enclosed by the given curves and find its area.