Volumes of Solids of Revolution Quiz

Volumes of Solids of Revolution

• Solids of revolution are solids whose shapes can be generated by revolving plane regions about axes.
• The method of slicing is commonly used to find the volumes of solids of revolution.
• The disk method, washer method, and cylindrical shell method can be used to find the volumes of solids of revolution.

Disk Method

• If a region bounded by a curve y = f(x), the x-axis, x = a, and x = b is revolved about the x-axis, a solid is generated.
• The typical cross-section of the solid perpendicular to the axis of revolution is a disk of radius f(x) and area A(x) = π[f(x)]².
• The solid's volume is the integral of A from x = a to x = b.

Exercises

• Find the volume of a cap of height h formed from a sphere of radius r.
• Find the volume of a right pyramid with a square base of side and height.
• Calculate the volume of the solid generated by revolving the plane region bounded by y = x, x = 4, and y = 1/2 about the x-axis.
• Compute the volume of the solid generated by revolving the plane region bounded by y = x², y = 9, and x = 0 about the x-axis.

Examples

• The base of a solid is an ellipse b²x² + a²y² = a²b². Each cross-section perpendicular to the x-axis is a square with ends of a side on the ellipse.
• A solid has a circular base of radius 2. Parallel cross-sections perpendicular to its base are equilateral triangles.
• The radius of a hemispherical vat is 5 ft, and it contains a liquid to a depth of 4 ft. Find the volume of the liquid.