Calculus I - Lecture 20: The Fundamental Theorem of
Calculus
Lecturer: Professor Nagpal
April 30, 2025
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Discussion up until this point:
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The Fundamental Theorem of Calculus
Part 1
Part 1 establishes the relationship between differentiation and integration and is stated as follows:
FTC PART 1: If f (x) is continuous over an interval [a, b], and the function F (x) is defined by
Z x
f (t) dt
F (x) =
a
then F ′ (x) = f (x) over [a, b].
Discussion:
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Comment on Notation:
Key Implications of Theorem:
Practice Problems: Use the FTC Part 1 to find the following derivatives of the following
functions:
Rx
• g(x) = 1 t31+1 dt
• g(r) =
Rr√
x2 + 4dx
0
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Part 2
Part 2 provides a practical way to evaluate definite integrals using antiderivatives and is stated
as follows:
FTC PART 2: If f (x) is continuous over the interval [a, b] and F (x) is any antiderivative of
f (x), then
Z b
f (x)dx = F (b) − F (a).
a
Discussion:
Notation:
Key Implications:
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Summary and Wrap-Up
We examined the FTC, learning how it links differentiation and integration. FTC Part 1 shows
they are inverse processes, while Part 2 offers a quick way to evaluate definite integrals. These
concepts are essential for calc 2.
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