What is Integration?
Integration is the process of finding the total accumulation of something , the area under a curve, the total distance traveled, the total work done. It is the reverse operation of differentiation and the second major pillar of calculus.
In this lesson
1 The Intuition: Adding Up Infinitely Many Pieces
Suppose you want to find the area under a curve y = f(x) between x=a and x=b. You could approximate it with rectangles: divide the interval into n thin slices, build a rectangle on each, and sum the areas. The more rectangles you use, the better the approximation.
Integration takes this to its limit: use infinitely many infinitely thin rectangles. Each has width dx (an infinitesimal) and height f(x), contributing f(x)dx to the sum. The integral symbol ∫ is an elongated S standing for "sum." The definite integral ∫ₐᵇ f(x)dx is the limit of this sum as the number of rectangles approaches infinity.
The formal definition: ∫ₐᵇ f(x)dx = lim(n→∞) Σᵢ f(xᵢ*)Δx. This is a sum of n rectangle areas, taken to its limit as n→∞ and Δx→0. This limit exists (and equals the integral) for any continuous function.
2 Definite Integrals and Area
∫ₐᵇ f(x)dx gives the signed area between f(x) and the x-axis from x=a to x=b. "Signed" means regions below the x-axis contribute negative area.
4.53 Indefinite Integrals and Antiderivatives
An antiderivative of f(x) is a function F(x) such that F'(x) = f(x). The indefinite integral ∫f(x)dx = F(x) + C, where C is the constant of integration (any constant has zero derivative, so there are infinitely many antiderivatives).
Example: ∫2x dx = x² + C, because d/dx(x²+C) = 2x. The +C is essential , without it, you're missing infinitely many valid answers.
4 Basic Integration Rules
5 Real-World Applications
Physics: if velocity is v(t), then total displacement = ∫v(t)dt. If acceleration is a(t), then velocity = ∫a(t)dt. Integration converts rates of change back into accumulated quantities.
Engineering: structural loads, electrical charge (integral of current), fluid flow, and heat transfer all involve integration. The center of mass of an irregular object is found by integration.
Probability: for continuous random variables, the probability of a range of outcomes is the integral of the probability density function. The normal distribution's probabilities come from integrating the bell curve formula.
Practice Problems
Sources & Further Reading
The explanations on this page draw on the following established sources. We link to primary and secondary sources so you can verify claims and go deeper on any topic.