Chapter 6  The Definite Integral"The need to calculate instantaneous rates of change led to the discoverers of calculus to an investigation of the slopes of tangent lines, and ultimately, to the derivative  to what we call differential calculus. In addition to a calculation method ( a "calculus") to describe how functions change at any given instant, they needed a method to describe how these instantaneous changes could accumulate over an interval to produce the function that describes the total change.
"Today, what we call the integral calculus or integration has two distinct interpretations. We begin this chapter by looking at integration as accumulation. But it also can be viewed as reversing the process of differentiation, what we call antidifferentiation. Newton's insight, that these two are connected, is what is called the Fundamental Theorem of Calculus." (Finney, Ross L., Franklin D. Demana, Bert K. Waits, and Daniel Kennedy. "Chapter 6." Calculus: Graphical, Numerical, Algebraic. 5th ed. N.p.: Pearson Education, n.d. 269. Print. AP Edition.) 
Chapter 6 reallife math connection.

Lesson 6.1  Estimating with Finite Sums
This lesson will teach students to estimate distance, areas, volumes, and accumulations using finite sums. Lesson 6.2  Definite Integrals
This lesson will teach students to interpret the definite integral as the limit of a Riemann sum and express the limit of a Riemann sum in integral notation as well as calculate definite integrals using areas. Lesson 6.3  Definite Integrals and Antiderivatives
This lesson will teach students to calculate a definite integral using areas and properties of definite integrals and apply definite integrals to problems involving the average value of a function. 
Lesson 6.4  Fundamental Theorem of Calculus
This lesson will teach students to analyze functions defined by an integral and evaluate definite integrals. Lesson 6.5  Trapezoidal Rule
This lesson will teach students to approximate definite integrals using the Trapezoidal Rule. 