What is an integral?
The integral reverses the derivative. It accumulates infinitely many infinitesimal pieces of a quantity — and the result is often a meaningful whole: area, distance, total cost.
The geometric idea
If the derivative answers "what's the rate?", the integral answers "what's the total?" The classic example: a velocity curve v(t). The area under that curve, between two times, is the distance traveled.
To compute that area, slice the time axis into many small intervals. On each interval, approximate the distance as v(t) · Δt. Sum the slices. The integral is the limit of that sum as the slices get infinitely thin.
The notation
∫ab f(x) dxRead as: "the integral of f from a to b." The ∫ is an elongated S for "sum." dx marks what variable is being sliced.
The result is a number — the signed area between the curve and the x-axis. "Signed" means area below the axis counts negative.
The Fundamental Theorem of Calculus
Differentiation and integration are inverse operations. If F is an antiderivative of f (so F′ = f), then:
∫ab f(x) dx = F(b) − F(a)This is what makes integrals computable. Without it, every integral would be a long Riemann sum. With it, you can often find an antiderivative in closed form and just subtract.
Worked example
Find ∫03 2x dx.
F(x) = x² (since d/dx x² = 2x) F(3) − F(0) = 9 − 0 = 9The signed area under y = 2x from x = 0 to x = 3 is 9. Geometrically, it's a triangle with base 3 and height 6, area = ½ · 3 · 6 = 9. ✓
Flashcard · Unit 4
Add 1 to the exponent, divide by the new exponent, add C.
Check your understanding
What is the antiderivative of f(x) = 3x²?