As before, indefinite integrals are presented first, stressing their importance for solving elementary differential equations. The rules for antiderivatives and the substitution method follow next. The list of available integrable functions now includes exponential functions and functions leading to logarithmic and inverse trigonometric functions. We no longer have to wait until Chapter 6, as in the standard edition, to integrate the tangent and cotangent functions. As in the previous edition, estimating with finite sums in a variety of application settings motivates the ideas of Riemann sums and definite integrals.
Students see the definite integral early as more than just a tool for finding area.
The section defining the definite integral as a limit of Riemann sums has been streamlined and now focuses on continuous functions. Piecewise-continuous functions are treated in the Additional Exercises at the end of the chapter. The treatment of volumes has been combined into two sections. Arc length formulas are developed for both explicit-function and parametric curves in the plane. Surface area has been moved to Chapter 13, where it is needed for surface integrals.