Differentiation asks: given a quantity, how fast is it changing? Integration asks the reverse: given a rate of change, what is the total accumulated quantity? These two questions form the twin pillars of calculus.
The integral sign ∫ represents summation taken to its limit—adding infinitely many infinitely small pieces. When you integrate a velocity function, you recover distance traveled. When you integrate a density function, you recover total mass. The pattern repeats across physics, economics, and probability: rates become totals through integration.
Two distinct but related concepts share this notation. The definite integral
∫abf(x)dx
computes a number—the signed area under a curve, the accumulated quantity between two bounds. The indefinite integral
∫f(x)dx
finds a function—the antiderivative whose derivative returns the original integrand. The Fundamental Theorem of Calculus reveals these as two faces of the same idea.
The Idea of Accumulation
Integration answers questions about totals. Given how fast something changes at each moment, what is the cumulative effect over an interval?
Consider velocity. If v(t) gives your speed at time t, then the integral
∫abv(t)dt
computes the total distance traveled from time a to time b. The function v(t) describes instantaneous rates; the integral accumulates those rates into a total.
This pattern appears throughout mathematics and science:
• Integrate force over distance to get work • Integrate density over volume to get mass • Integrate a probability density to get probability • Integrate marginal cost to get total cost
The integral formalizes "continuous summation"—adding contributions that vary smoothly rather than in discrete chunks.
Two Types of Integrals
The integral symbol carries two distinct meanings depending on context.
Definite Integrals
∫abf(x)dx
The limits a and b bound the region of integration. The result is a number representing accumulated quantity—area, volume, total change, or another aggregate measure.
Indefinite Integrals
∫f(x)dx=F(x)+C
No limits appear. The result is a family of functions—all antiderivatives of f(x), differing by an arbitrary constant C.
The same symbol serves both purposes. Context determines which interpretation applies: limits present means definite, limits absent means indefinite.
Notation and Terminology
The integral notation carries specific meaning in each component.
The integral sign ∫ is an elongated S, standing for "sum." It originated with Leibniz, who conceived integration as summing infinitesimal pieces.
The integrand f(x) is the function being integrated. It describes what is accumulated at each point.
The differential dx indicates the variable of integration and represents an infinitesimal width. Together, f(x)dx represents an infinitesimal contribution to the total.
For definite integrals, the limits of integration a and b specify where accumulation begins and ends:
∫abf(x)dx
The lower limit a appears at the bottom of the integral sign, the upper limit b at the top.
Definite Integrals
The definite integral computes signed area—the area between a curve and the x-axis, with regions below the axis counted as negative.
∫abf(x)dx
Positive where f(x)>0, negative where f(x)<0. The integral sums these signed contributions.
Rigorously, the definite integral arises as a limit of Riemann sums. Partition the interval [a,b] into subintervals, approximate the area with rectangles, and take the limit as the rectangles become infinitely thin. This construction gives precise meaning to "area under a curve."
The indefinite integral reverses differentiation. If F′(x)=f(x), then F is an antiderivative of f, and we write:
∫f(x)dx=F(x)+C
The constant C is essential. Since the derivative of any constant is zero, infinitely many functions share the same derivative. The "+C" represents this entire family.
Finding antiderivatives is the core skill of indefinite integration. Unlike differentiation, which follows systematic rules, integration often requires insight, technique, and pattern recognition.
Not every function yields to direct antidifferentiation. Integration techniques provide methods for transforming difficult integrals into manageable ones.
Substitution reverses the chain rule. Identify an inner function and its derivative, change variables, and integrate.
Integration by parts reverses the product rule:
∫udv=uv−∫vdu
Partial fractions decompose rational functions into simpler pieces. Trigonometric substitution handles integrals involving square roots of quadratics.
Choosing the right technique is an acquired skill—pattern recognition developed through practice.
Standard definite integrals require finite intervals and bounded integrands. Improper integrals extend integration beyond these constraints.
Integrals over infinite intervals:
∫1∞x21dx=b→∞lim∫1bx21dx=1
Integrals of unbounded functions:
∫01x1dx=a→0+lim∫a1x1dx=2
An improper integral converges if the limit exists and is finite; otherwise it diverges. The distinction matters—some infinite regions have finite area, others do not.
Evaluating integrals combines all the preceding ideas: recognize the form, apply appropriate techniques, and verify the result.
Start with direct antidifferentiation. If the integrand matches a known form, apply the formula. If not, consider substitution, parts, or other techniques.
For definite integrals, set up the problem correctly: identify bounds, express the integrand in terms of the integration variable, and apply the Fundamental Theorem.
Verification is simple: differentiate your answer. The derivative of the antiderivative should return the original integrand. This check catches algebraic errors and sign mistakes.