Lesson 7: 6.7 The Fundamental Theorem of Calculus and Definite Integrals

How can we find the exact area under a curve without using Riemann sums or geometric formulas?
What is the relationship between the boundaries of an integral (a and b) and the values of the antiderivative F(x)?
Does the constant of integration (+C) matter when evaluating a definite integral? Why or why not?
How does the FTC link the "rate of change" of a quantity to its "total change"?

Fundamental Theorem of Calculus (Part 2)

Evaluation Theorem

Antiderivative

Net Change

Upper Limit / Lower Limit

College Board AP Calculus CED: FUN-6.B (Evaluate definite integrals using the Fundamental Theorem of Calculus).

Mathematical Practices: MP 1 (Implementing Mathematical Processes), MP 2 (Connecting Representations).

Lesson Description:
Students learn the mechanical process of evaluating integrals by finding the antiderivative F(x) and calculating the difference F(b) - F(a).
The lesson transitions from the conceptual "accumulation" of previous sections to the algebraic efficiency of the Evaluation Theorem. Students practice with power functions, trigonometric functions, and exponential functions, while learning how to properly use evaluation notation (the vertical bar or brackets).
The purpose of 6.7 is to provide students with the primary tool used for the remainder of the course. While Riemann sums are used for approximation, the FTC Part 2 allows for exact calculation. This lesson is essential for solving "Net Change" problems, where students must find the total displacement of a particle or the total amount of water that has entered a tank given a rate function.

DOK 1 (Recall): Evaluating basic definite integrals of polynomial functions.

DOK 2 (Skill/Concept): Applying the FTC to functions requiring trigonometric or transcendental antiderivatives.

DOK 3 (Strategic Thinking): Solving for an unknown boundary or constant within a definite integral equation

Support (Scaffolding):The +C Cancellation Proof: Show students the algebra of (F(b) + C) - (F(a) + C) to visually demonstrate why the constant of integration cancels out in definite integrals.
Subtraction Reminders: Use a template that provides empty parentheses to ensure students evaluate the upper limit first and subtract the entire lower limit evaluation (especially important for negative terms).
Antiderivative Reference: Provide a "Refresher Sheet" for basic integration rules so they can focus on the evaluation process.

Extension (Enrichment):Absolute Value Integrals: Challenge students to evaluate integrals by identifying where the function changes sign and breaking the integral into two pieces.
Geometric Verification: Have students evaluate an integral algebraically and then verify it using the area of a triangle to build confidence in the theorem.

Ap College Board Assessments