Applications of integrals

An integral adds up many small pieces. In this chapter, we use that idea to find areas between curves, volumes of solids, and average values.

Each problem follows the same plan: cut the quantity into thin pieces, find the size of one piece, and add the pieces with an integral.

This chapter covers the following topics:

Areas between curves

The area between two curves: top minus bottom, curves that cross, and integrating with respect to y.

Volumes by slicing

Volumes from cross-sections, the disk and washer methods for solids of revolution, and where the sphere and cone formulas come from.

Volumes by cylindrical shells

The shell method for solids of revolution, and how to choose between shells and washers.

Average value of a function

The average of a function over an interval, the Mean Value Theorem for Integrals, and examples with temperature and velocity.

Practice questions

Practice problems on every section of this chapter, with full step-by-step solutions.

Before you start

This chapter uses the integrals from the last chapter. These pages may help:

• The Fundamental Theorem of Calculus
• The substitution rule
• Areas and Riemann sums (the idea of slicing)
• Curve sketching (drawing the region first)