Integrals

The integral is the second big idea of calculus. It starts with a question about area: how big is a region with a curved side? The answer is a limit of sums of thin rectangles.

In this chapter, you will see that integrals and derivatives undo each other. This is the Fundamental Theorem of Calculus. It turns hard limits into quick calculations, and it lets us find totals from rates of change.

This chapter covers the following topics:

Antiderivatives

Running differentiation backward, why all antiderivatives differ by a constant, antiderivative formulas, and finding position from acceleration.

Areas and Riemann sums

Estimating areas with rectangles, sigma notation, the exact area as a limit, and distance from velocity.

The definite integral

The definition of the definite integral, area and net area, and the properties of integrals.

The Fundamental Theorem of Calculus

How derivatives and integrals undo each other, and the fast way to compute definite integrals.

Indefinite integrals and net change

Integral notation for antiderivatives, a table of integrals, the net change theorem, and displacement versus distance.

The substitution rule

The chain rule backward: how to choose u, definite integrals with new limits, and even and odd functions.

Before you start

This chapter uses derivatives all the time. These pages may help:

• Differentiation rules (especially the chain rule)
• The Mean Value Theorem (functions with the same derivative)
• Limits at infinity (used for exact areas)
• Rational exponents