Calculus I · Unit 3A · lesson
Fundamental Theorem of Calculus, Part II
Learn Fundamental Theorem of Calculus, Part II through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Section overview
The Fundamental Theorem of CalculusWhat this section is building
Learn Fundamental Theorem of Calculus, Part II through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Differentiating a running total recovers its current integrand, while evaluating an accumulated total subtracts antiderivative endpoint values.
Separate FTC Part I, FTC Part II, net change, and variable-bound chain-rule tasks before manipulating notation.
Forgetting a chain-rule factor at a variable bound, reversing endpoint subtraction, or adding +C to a definite value.
Learning objectives
Evaluate definite integrals with antiderivatives and connect the result to net accumulated change.
Fundamental Theorem of Calculus, Part II
Why antiderivatives evaluate exact accumulation
Riemann sums define the definite integral, but evaluating a limit of sums directly would be unbearable for most functions. The second part of the Fundamental Theorem provides the computational shortcut: if , then . The endpoint difference captures all of the tiny accumulated contributions at once.
The notation should be read as an instruction to substitute both endpoints and subtract in the correct order. The constant of integration cancels, which is why definite integrals do not need . Even so, the theorem is not merely a formula to memorize. It depends on recognizing an antiderivative and on understanding that the result is net accumulation, with sign and units inherited from the original integrand and variable.
FTC Part II
If is continuous on and , then
The theorem is astonishing because it turns a limit of many sums into two endpoint evaluations.
Evaluate a definite integral
The bracket notation means evaluate the antiderivative at the upper endpoint and subtract its value at the lower endpoint.
u3a-ftc2-01Evaluate .
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An antiderivative is .
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The definite integral is not , and the lower endpoint is not optional. Write the bracket step explicitly until endpoint subtraction is automatic.
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