The Definite Integral
“A definite integral squeezes an area down to a single number.”
The formula
∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a)How to read it: the area from a to b is the antiderivative F evaluated at b minus at a
- a, b
- — the start and end of the interval (lower and upper limits)
- F
- — an antiderivative of f (a function whose derivative is f)
- F(b) − F(a)
- — the value at the top minus the value at the bottom
The hook
A definite integral needs no 'infinite adding' — you just plug two numbers into an antiderivative and subtract.
In plain words
The area over [a,b], computed not by a Riemann sum but by finding an antiderivative F and taking F(b)−F(a).
The intuition
A definite integral is a single number (not a function). Like mile-markers, the difference between the amount piled up to the end, F(b), and to the start, F(a), is the new area accumulated across that stretch.
How it's built
① Find an antiderivative F of f (a function that differentiates back to f), ② plug in the top b, and ③ subtract the value at the bottom a. The bracket [F(x)]ₐᵇ denotes that 'plug and subtract'.
Example
∫₀³ x² dx: the antiderivative is x³/3. [x³/3]₀³ = 27/3 − 0 = 9.
Common mistake
A definite integral is a number, not a function. And it's 'signed area', so parts below the x-axis count as negative — the value can be zero or even negative.
Where it's used
Pulling out an actual number for a 'total across an interval' — total distance, an average value, work, probability, and so on.
Where it came from
Leibniz's ∫ symbol is a stretched S, from the Latin summa ('sum') — the trace of 'adding' still lives in the notation.
Prerequisites
Quick check
What is the value of ∫₀² x dx?
- 1
- 2✓
- 4
- 0
Practice
Evaluate ∫₀² 2x dx.
Answer: 4
- Antiderivative: x²
- [x²]₀² = 4 − 0 = 4
Takeaway: Find the antiderivative, plug in top and bottom, subtract.
Evaluate ∫₁² 3x² dx.
Answer: 7
- Antiderivative: x³
- [x³]₁² = 8 − 1 = 7
Takeaway: Thanks to the coefficient 3, the antiderivative is a clean x³.
Evaluate ∫₀¹ (2x+1) dx.
Answer: 2
- Antiderivative: x²+x
- [x²+x]₀¹ = (1+1) − 0 = 2
Takeaway: Find the antiderivative term by term and combine.