What an Integral Measures: Area, Accumulation and the Fundamental Theorem

An integral adds up infinitely many tiny pieces to produce one total. The most familiar use is finding the area under a curve, but the same idea measures distance travelled from a speed graph, the total water that flows through a pipe, and the total charge stored by a capacitor. This guide explains the definite integral through Riemann sums, shows how the fundamental theorem of calculus turns the sum into a simple subtraction, and gives worked examples and a table of standard results.

Area as a limit of rectangles

Suppose you want the area between the graph of f(x) = x² and the horizontal axis, from x = 0 to x = 1. No standard rectangle or triangle formula covers a curved boundary, so you approximate it. Divide the interval into n equal strips, each of width Δx = 1/n. Over each strip, place a rectangle whose height matches the function at one chosen point.

Using the right endpoint of each strip, the rectangles have heights (k/n)² for k from 1 to n. Each area is height times width, so the total is a sum called a Riemann sum. For n = 4, the heights are 1/16, 4/16, 9/16, and 16/16, and their sum is 30/16. Multiplying by the width 1/4 gives 30/64, which is about 0.469. That overestimates the true area, because the rectangles stick above the curve.

Taking the limit

The approximation improves as n grows. The exact area is the limit of the Riemann sum as the number of strips goes to infinity. This limit is written with the integral sign:

∫ from 0 to 1 of x² dx

The symbol ∫ is an elongated S, standing for sum. The numbers 0 and 1 are the limits of integration, and dx marks the variable and the width of each strip. For x², the limit equals 1/3, which is about 0.333. The estimate with four strips was far too large, and with a thousand strips it would be very close to this value.

Definite and indefinite integrals

A definite integral has limits and produces a number. It answers the question of how much accumulates between two points.

An indefinite integral has no limits and produces a family of functions. It is the reverse operation of differentiation. The indefinite integral of 2x is x² + C, where C is an arbitrary constant. The constant appears because every function with the same derivative differs from the others only by a fixed value.

The fundamental theorem of calculus

The most important result in the subject links these two ideas. The fundamental theorem of calculus states two linked facts.

First, if F is an antiderivative of f, meaning F' = f, then the definite integral of f from a to b equals F(b) - F(a). This turns a limit of sums into the evaluation of a function at two points.

Second, the function defined by accumulating f from a fixed starting point has derivative equal to f itself. In short, integration and differentiation undo each other.

Using the theorem on the earlier example, an antiderivative of x² is x³/3. Then the integral from 0 to 1 equals (1³/3) - (0³/3) = 1/3. This matches the limit computed from the Riemann sums.

Worked examples

Consider the integral of 3x² from 0 to 2. An antiderivative is x³, so the value is 2³ - 0³ = 8. This is the area under the parabola y = 3x² over the first two units.

Consider the integral of cos x from 0 to π/2. An antiderivative is sin x, so the value is sin(π/2) - sin 0 = 1 - 0 = 1. The area under one quarter of a cosine wave equals exactly 1.

Consider a car whose speed is v(t) = 4t metres per second for 0 ≤ t ≤ 5. The distance travelled is the integral of v from 0 to 5, which is 2t² evaluated from 0 to 5, giving 50 metres. Here the integral measures accumulated distance rather than area, yet the mathematics is identical.

Properties that simplify work

Several properties make integrals easier to manipulate. The integral of a sum is the sum of the integrals, and a constant factor can be pulled outside. The integral from a to b is the negative of the integral from b to a. Splitting the interval also works: the integral from a to c equals the integral from a to b plus the integral from b to c.

Symmetry can save effort too. If a function is odd, meaning f(-x) = -f(x), then its integral over an interval symmetric about zero is zero, because the positive and negative areas cancel.

Table of standard integrals

The table below lists the integrals used most often. Each formula can be checked by differentiating the right side and confirming that it returns the integrand.

Function	Antiderivative	Typical meaning
xⁿ (n ≠ -1)	xⁿ⁺¹ / (n + 1)	Accumulated quantity from a power rate
1/x	ln |x|	Logarithmic accumulation
eˣ	eˣ	Total from exponential growth
sin x	-cos x	Net displacement from a sine velocity
cos x	sin x	Net displacement from a cosine velocity

Every antiderivative also carries an arbitrary constant C, which disappears when a definite integral subtracts two values.

Common mistakes

A frequent error is forgetting that a definite integral depends on its limits. The result is a number, so do not leave the variable in the answer.

Another mistake is omitting the constant of integration in indefinite integrals. Without C, the answer describes only one member of the family of antiderivatives, which is incomplete.

A third issue is ignoring signs. A definite integral counts area below the axis as negative. When the question asks for total area without sign, split the integral at the zeros of the function and take absolute values of each piece.

Why integrals matter

Integrals appear wherever a total must be built from a varying rate. Engineers compute the work done by a changing force, statisticians compute probabilities as areas under density curves, and economists compute total consumer surplus from demand curves. The common thread is that a small contribution is added over a continuous range.

The Riemann sum gives an intuitive picture of this process, and the fundamental theorem gives an efficient method of calculation. Using both together makes the subject feel like one coherent idea rather than a list of techniques.

Key takeaways

A definite integral is the limit of a Riemann sum. It measures the total accumulated by a function over an interval, whether that total is geometric area or a physical quantity such as distance.

The fundamental theorem of calculus says that the integral equals the difference of an antiderivative at the endpoints. This converts a limit of sums into a short computation.

Check each antiderivative by differentiating it, and keep track of limits and signs. These habits prevent most errors in beginning integration.