The Definite Integral and the Fundamental Theorem of Calculus
Simple Explanation
The Fundamental Theorem of Calculus (FTC) says that a definite integral β«βα΅f(x)dx can be evaluated by finding any antiderivative F of f, and computing F(b)βF(a) β turning a quantity originally defined as an infinite sum of infinitesimal slices into a simple two-value calculation.
Why Do We Need It?
This theorem is the bridge between differentiation and integration β without it, evaluating a definite integral exactly would require the much harder machinery of limits of Riemann sums every single time.
See It
A shaded trapezoid region under a straight rising line, bounded on the left by x=1 and on the right by x=3, sitting above the x-axis
Formula
The Fundamental Theorem of Calculus
β«βα΅ f(x) dx = F(b) β F(a), where F'(x) = f(x)
Connects the two central operations of calculus β differentiation and integration β by showing that a definite integral can be evaluated using any antiderivative, without needing to sum infinitely many slices directly.
- a, b
- β the lower and upper limits of integration
- F
- β any antiderivative of f (F'(x)=f(x))
When to use it: Whenever a definite integral needs to be evaluated exactly, given a function with a known antiderivative.
Worked Example
Evaluate a definite integral using the Fundamental Theorem of Calculus
Evaluate β«βΒ³ 2x dx.
Why Does This Work?
This shaded trapezoid has area (2+6)/2 Γ (3β1) = 8 by the elementary trapezoid-area formula β exactly matching the integral's value, confirming that the Fundamental Theorem's shortcut (F(b)βF(a)) really does compute the true area under the curve, without needing to sum infinitely many thin slices by hand.
Real-Life Example
Total displacement from a velocity function
A vehicle's velocity is recorded as a function of time, and the total distance traveled between two specific times is needed.
The Fundamental Theorem of Calculus lets that total distance be computed exactly, as F(b)βF(a) for any antiderivative F of the velocity function, without needing to sum an enormous number of tiny time-slices.
Practice
Evaluate β«βΒ² 6xΒ² dx.
MediumCommon mistake
Computing F(a)βF(b) instead of F(b)βF(a) β subtracting in the wrong order flips the sign of the final answer.
Quick Review
- β«βα΅ f(x)dx = F(b) β F(a), for any antiderivative F of f.
- This turns a definite integral into a simple two-value subtraction.
- Always evaluate at the UPPER limit first, then subtract the value at the lower limit.