Definition: The definite integral of a function from a to b is the net signed area between its graph and the x-axis, found as the limit of Riemann sums.
What is a definite integral?
The expression ∫ from a to b of f(x) dx measures net area: regions above the x-axis count positive, regions below count negative. It is the limit of Riemann sums as the rectangles get infinitely thin.
How do you compute it?
Find an antiderivative F, then compute F(b) – F(a).
Worked example: ∫ from 1 to 4 of (3x2 – 2x) dx. An antiderivative is F(x) = x3 – x2. Then F(4) = 64 – 16 = 48 and F(1) = 1 – 1 = 0. The integral equals 48.
Useful properties: reversing the limits flips the sign, and splitting the interval adds the pieces.
What mistake is typical?
Treating net area as total area. The integral of sin x from 0 to 2 pi is 0 because the positive and negative parts cancel. Mathbot splits the interval at each x-intercept when total area is requested.
Where to learn it
- Integrals, Riemann Sums and Accumulation – lesson
- Applications of Integration: Area and Volume – lesson