The Fundamental Theorem of Calculus and Definite Integrals
Check an antiderivative and continuity, then evaluate signed accumulation as an endpoint difference.
College Board Topic 6.7 · CED p. 119 (PDF p. 124) · CED effective Fall 2020; project course outline 2026–27.
An antiderivative F of f satisfies F′=f. If f is continuous on [a,b], the Fundamental Theorem evaluates the definite integral as F(b)−F(a). This connects accumulated change to the difference in an antiderivative’s endpoint values. An accumulation function ∫ₐˣ f(t)dt is itself one antiderivative when f is continuous.
Write the bounds on your antiderivative and subtract in the order upper minus lower. The arbitrary constant cancels, so a definite answer has no +C. If an integrand changes sign, this still gives net signed area. For total positive area, split at zeros and change signs on negative pieces.
Let's work through it
- For ∫₀² (3x²−4x)dx, use F(x)=x³−2x². Reason: Differentiating F recovers the continuous integrand.
- Compute F(2)−F(0)=8−8=0. Reason: The Fundamental Theorem evaluates net change; cancellation is allowed.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Why is there no +C in the final value of a definite integral?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Why is there no +C in the final value of a definite integral?
Worked answer and reason
It cancels in (F(b)+C)−(F(a)+C).
Both endpoints use the same antiderivative constant.
Retrieval 2
Is an integral of zero enough to prove f is identically zero?
Worked answer and reason
No.
Positive and negative contributions can cancel.
Catch the tempting mistake
- F(b)−F(a) always gives total geometric area.
- It gives signed accumulation. Use ∫|f|, or separate positive and negative pieces, for total area.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 6 problem 17 · No calculator · analytic
- Unit 6 problem 18 · No calculator · analytic
- Unit 6 problem 30 · No calculator · analytic
For the teacher
- Require a derivative check on a proposed antiderivative. Keep the continuity condition visible when students invoke the theorem.
Glossary and deeper reading · 3 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
-
Antiderivative
An antiderivative of a function is another function whose derivative equals the original function on an interval. Two antiderivatives on the same interval differ by a constant. For example, x² is an antiderivative of 2x, and x²+7 is another. An antiderivative is a function, whereas a definite integral is an accumulated value.
-
Definite integral
A definite integral accumulates a function over a specified interval as a limit of Riemann sums when that limit exists. Its sign includes the function’s sign and the orientation of the limits. For example, the integral of the constant −2 from 0 to 3 is −6, although the geometric area is 6. In a rate model, the result has rate units multiplied by input units.
-
Integrand
The integrand is the function or expression being integrated. The differential identifies the variable of integration. For example, in the integral of x² with respect to x, the integrand is x². Bounds belong to the integral’s interval rather than to the integrand itself.