Lesson 4: 6.4 The Fundamental Theorem of Calculus and Accumulation Functions

What happens when the upper limit of a definite integral is a variable instead of a constant?
How does the "area so far" change as we move along the x-axis?
What is the mathematical relationship between the rate of change of an accumulation function and the original integrand?
How do we apply the Chain Rule when the upper limit of integration is a function, such as g(x), rather than just x?

Accumulation Function
Fundamental Theorem of Calculus (Part 1)
Dummy Variable
Variable Limit of Integration
Derivative of an Integral

College Board AP Calculus CED: FUN-5.A (Represent accumulation functions using definite integrals), FUN-5.A.1 (The FTC states the connection between differentiation and integration).

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

In this lesson, students move from calculating static areas to analyzing "functions defined by integrals." The lesson's climax is the Fundamental Theorem of Calculus. Students practice using this "shortcut" to find derivatives of complex integral expressions and learn how to handle cases where the upper limit is a composition of functions (requiring the Chain Rule).

The purpose of 6.4 is to reveal the inverse relationship between differentiation and integration. This is the "glue" of calculus. It shifts the student's perspective from viewing integration as a geometry problem to viewing it as an algebraic operation. This concept is vital for FRQs where students are given a graph of f and asked to find the properties (extrema, concavity) of its integral g(x).

DOK 1 (Recall): Evaluating a simple derivative of an integral where the upper limit is x.
DOK 2 (Skill/Concept): Using the Chain Rule to find intergal.
DOK 3 (Strategic Thinking): Analyzing a graph of f(t) to determine where its accumulation function F(x) is increasing, decreasing, or has a point of inflection.

Support (Scaffolding):"The Eraser" Analogy: Explain the FTC as an "eraser" where the derivative sign and the integral sign cancel each other out, leaving only the function behind (remind them to "plug in" the limit).
Dummy Variable Clarification: Use a color-coded system to show that t is just a placeholder for the horizontal axis, while x is the "stop sign" for the area.
Extension (Enrichment):Both Limits are Variables: Challenge students to find the derivative of both integrals by splitting the integral at a constant.

AP College Board Assessment