What Is an Integral? The Mathematics of Accumulation

An integral adds up infinitely many infinitely small pieces to get a finite total.

That sounds impossible. It's not. It's the most powerful accounting trick in mathematics.

Here's the unlock: you already understand integration intuitively. When you estimate the area of an irregular shape by covering it with tiny squares and counting them, you're doing a crude version of what integrals do perfectly. The integral is what happens when those squares become infinitely small — and there are infinitely many of them.

Derivatives tell you how fast something changes. Integrals tell you how much has accumulated. Your speedometer is a derivative (instantaneous rate). Your odometer is an integral (total distance). Same underlying motion, opposite questions.

The Problem: Area Under a Curve

How do you find the area of a shape that isn't a rectangle, triangle, or circle?

Geometry gives formulas for regular shapes. But what about the area under a curve like y = x²? There's no simple formula for "parabola-shaped region."

The ancient Greeks struggled with this. Archimedes came close, using a method of exhaustion — filling curved regions with triangles. But it was case-by-case, heroic mathematics. Each new curve required new ingenuity.

Calculus solved this once and for all.

The Solution: Slice It Infinitely Thin

Divide the region into vertical rectangles.

Each rectangle has width Δx (a small change in x) and height f(x) (the function's value at that point). The area of each rectangle is height × width = f(x) · Δx.

Add up all the rectangles. You get an approximation of the area.

Now make the rectangles thinner. More of them, each with smaller Δx. The approximation gets better — the rectangles fit the curve more tightly.

The integral is the limit of this process as the rectangles become infinitely thin (Δx → 0) and infinitely numerous.

The result isn't an approximation. It's exact.

The Notation

The integral of f(x) from a to b is written:

∫ₐᵇ f(x) dx

Here's what each piece means:

• ∫ — the integral sign, an elongated S (for "summa," Latin for sum)
• a and b — the bounds: where we start and stop summing
• f(x) — the function whose values we're adding up
• dx — a reminder that we're summing in the x-direction, taking infinitely small slices of width dx

The whole expression reads: "the integral from a to b of f(x) with respect to x."

Leibniz designed this notation to look like a sum — because that's what it is. Just an infinite one.

The Geometric Meaning: Area Under the Curve

For a positive function f(x), the integral from a to b equals the area between the curve, the x-axis, and the vertical lines x = a and x = b.

This is the most common interpretation, but it's not the only one.

If f(x) goes negative, the integral subtracts that area instead of adding it. Areas below the x-axis count as negative.

This makes physical sense. If f(x) represents velocity and you integrate it, you get displacement — which can be positive or negative depending on direction.

The Physical Meaning: Accumulation

Forget area. Think accumulation.

If f(t) represents how fast something is changing (a rate), then the integral of f(t) represents how much total change has occurred.

Rate × Time = Amount

But that formula only works when the rate is constant. When the rate varies, you need calculus.

The integral generalizes "rate × time" to handle any rate function, no matter how it varies. It adds up all the infinitesimal contributions f(t)·dt.

Examples:

• Velocity (rate of position change) integrates to displacement
• Power (rate of energy transfer) integrates to total energy
• Current (rate of charge flow) integrates to total charge
• Marginal cost (rate of cost increase) integrates to total cost

Whenever you have a rate and want the total, you integrate.

Riemann Sums: The Formal Definition

A Riemann sum approximates an integral by adding up rectangle areas.

Divide the interval [a, b] into n subintervals. In each subinterval, pick a point xᵢ and form a rectangle with height f(xᵢ) and width Δx = (b-a)/n.

The Riemann sum is:

Σᵢ f(xᵢ) · Δx

This is just adding up rectangle areas. Nothing infinite yet.

The integral is the limit of Riemann sums as n → ∞ (equivalently, as Δx → 0):

∫ₐᵇ f(x) dx = lim[n→∞] Σᵢ f(xᵢ) · Δx

If this limit exists and equals the same value regardless of how you choose the xᵢ points, the function is called integrable.

Most functions you encounter in practice are integrable. The formal conditions (bounded, continuous except at finitely many points) are almost always satisfied.

Left, Right, and Midpoint

For any Riemann sum, you must choose where in each subinterval to evaluate f(x).

Left Riemann sum: Use the left endpoint of each subinterval. Right Riemann sum: Use the right endpoint. Midpoint rule: Use the center of each subinterval.

For increasing functions, left sums underestimate, right sums overestimate. For decreasing functions, it's reversed.

As n → ∞, all three approaches converge to the same answer: the true integral.

Why It Works: The Magic of Limits

Here's the mind-bending part: infinity makes the approximation exact.

With finitely many rectangles, you have errors — gaps between rectangles and curve, or overlaps. The sum isn't equal to the area; it's close.

But as rectangles become infinitely thin, those errors become infinitely small. And infinitely many infinitely small errors can sum to zero.

This is the genius of calculus. It tames infinity — uses it constructively instead of letting it cause paradoxes.

The limit exists because the errors shrink faster than the number of rectangles grows. The total error goes to zero even though you're adding infinitely many pieces.

A Concrete Example

Find the area under y = x from x = 0 to x = 2.

This region is a triangle with base 2 and height 2, so the area should be ½ · 2 · 2 = 2.

Let's verify with Riemann sums.

Divide [0, 2] into n equal pieces, each with width Δx = 2/n. Using right endpoints:

• First rectangle: height = 2/n, width = 2/n, area = 4/n²
• Second rectangle: height = 4/n, width = 2/n, area = 8/n²
• ...
• k-th rectangle: height = 2k/n, width = 2/n, area = 4k/n²

Total sum = Σₖ 4k/n² = (4/n²) · Σₖ k = (4/n²) · n(n+1)/2 = 2(n+1)/n

As n → ∞: 2(n+1)/n → 2

The integral equals 2, confirming the triangle formula.

The Integral as an Operator

Think of integration not just as area, but as an operation — something you do to functions.

Input: a function f(x) Output: another function F(x) (the antiderivative) or a number (the definite integral)

This operator perspective becomes crucial in the Fundamental Theorem of Calculus: differentiation and integration are inverse operations.

Just as addition and subtraction undo each other, just as multiplication and division undo each other, differentiation and integration undo each other.

Take a derivative, then integrate: you get back where you started (up to a constant). Integrate, then take a derivative: you get back where you started (exactly).

That's the big reveal of the next article.

Why This Matters

Integration isn't just abstract mathematics. It's the language of totals.

Any quantity that accumulates from a varying rate requires integration:

• Distance from velocity
• Work from force
• Probability from probability density
• Present value from cash flows
• Total heat from temperature changes

The derivative asks "how fast?" The integral asks "how much total?" Every phenomenon involves both questions. Calculus gives you both answers.

Further Reading

• Stewart, J. Calculus: Early Transcendentals. The standard undergraduate text.
• Thompson, S. P. Calculus Made Easy. The classic plain-English introduction.
• Strang, G. Calculus. Free MIT textbook with excellent explanations.

This is Part 1 of the Integrals series. Next: "The Fundamental Theorem of Calculus" — why differentiation and integration are opposite operations.

Part 1 of the Calculus Integrals series.

Previous: Integrals Explained Next: The Fundamental Theorem of Calculus: Derivatives and Integrals Are Opposites