What Is an Integral? A Beginner-Friendly Explanation

Integration as adding up

If a car drives at a constant 60 km per hour for 2 hours, it travels $60 \times 2 = 120$ km. On a graph of speed against time, that distance is the area of a rectangle of height 60 and width 2. When the speed changes, the region under the graph is no longer a rectangle, but the distance is still the area under the speed curve. Finding that area is what an integral does.

More generally, whenever a quantity is a rate multiplied by an amount, such as speed times time, density times volume or force times distance, and the rate varies, the total is an integral. The derivative breaks a total into rates; the integral rebuilds the total from the rates.

Riemann sums: approximating with rectangles

To find the area under $f(x) = x^2$ from $x = 0$ to $x = 2$, split the interval into 4 strips of width $0.5$. Using the right-hand edge of each strip for the height gives rectangles with heights $f(0.5)$, $f(1)$, $f(1.5)$ and $f(2)$:

Using the left-hand edges instead gives $0.5\,(0 + 0.25 + 1 + 2.25) = 1.75$. The true area lies between the two. As the number of strips $n$ grows, both sums approach the same value, and that limit is the definite integral:

The integral sign is a stretched S for sum, $f(x)$ is the height, and $dx$ represents the infinitely thin width. For $x^2$ on $[0, 2]$ the limit is exactly $\frac{8}{3} \approx 2.667$. The idea of a limit, explained in our limits guide, is what turns an approximation into an exact value.

Signed area

Where the graph lies below the x-axis, $f(x)$ is negative, so the rectangles contribute negative amounts. The definite integral therefore measures signed area: area above the axis minus area below it.

For example, $\int_0^{2\pi}\sin x\,dx = 0$, because the hump above the axis between 0 and $\pi$ exactly cancels the one below between $\pi$ and $2\pi$. Each hump has area 2, so the total geometric area is 4. To find total area, split the interval where the function changes sign and add the absolute values of the pieces.

Antiderivatives and the indefinite integral

An antiderivative of $f$ is a function $F$ whose derivative is $f$. Since $\frac{d}{dx}x^3 = 3x^2$, the function $x^3$ is an antiderivative of $3x^2$. So is $x^3 + 7$, or $x^3$ plus any constant, because constants have derivative zero. The indefinite integral collects them all:

Each basic derivative rule, read backwards, gives an integration rule:

The Fundamental Theorem of Calculus

The two meanings of integral, an area found by a limit of sums and a family of antiderivatives, are connected by the Fundamental Theorem of Calculus. Its practical form says that if $F$ is any antiderivative of a continuous function $f$, then:

This is remarkable: an infinite sum of rectangles reduces to two function values. The other part of the theorem says that the accumulated area function $A(x) = \int_a^x f(t)\,dt$ has derivative $f(x)$, so integration and differentiation undo each other.

The difference between the two kinds of integral is explored further in definite vs indefinite integrals.

Two techniques beyond the basic rules

Integration has no product or chain rule that works mechanically every time, so some techniques reverse those derivative rules.

Applications

• Distance from velocity: if $v(t) = 3t^2$ m/s, the displacement over the first 2 seconds is $\int_0^2 3t^2\,dt = 8$ m.
• Area between curves: between $y = x$ and $y = x^2$ on $[0, 1]$ the area is $\int_0^1 (x - x^2)\,dx = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$.
• Average value: the average of $f$ on $[a, b]$ is $\frac{1}{b - a}\int_a^b f(x)\,dx$.
• Probability: areas under a probability density curve give probabilities, as with the normal curve behind z-scores.
• Volumes, work done by a variable force, total cost from marginal cost and centres of mass are all integrals.

Common mistakes

• Forgetting $+ C$ on an indefinite integral.
• Applying the power rule to $\frac{1}{x}$, which would divide by zero; its integral is $\ln|x|$.
• Thinking $\int f g\,dx$ equals $\int f\,dx \cdot \int g\,dx$.
• Reading a definite integral as total area when part of the graph is below the axis.
• Computing $F(a) - F(b)$ instead of $F(b) - F(a)$.

The integral calculator finds antiderivatives with each step shown, and the definite integral calculator evaluates integrals over an interval and reports the numerical value.

Why the Fundamental Theorem makes sense

Return to the car. Suppose $s(t)$ is its position and $v(t) = s'(t)$ its velocity. Adding up velocity times tiny time intervals over a journey gives the total change in position, which is the final position minus the starting position. In symbols, $\int_a^b v(t)\,dt = s(b) - s(a)$, which is exactly the theorem with $s$ playing the role of the antiderivative.

Seen this way, the theorem is almost common sense: adding up all the small changes in a quantity gives its total change. What makes it powerful is that it turns hard limit calculations into easy subtraction, as long as an antiderivative can be found.

Numerical integration

When no antiderivative can be written down, or when the function is only known from measurements, integrals are estimated numerically. Better estimates come from better shapes than rectangles. The trapezoidal rule averages the left and right sums; for $x^2$ on $[0, 2]$ with 4 strips it gives $\frac{1.75 + 3.75}{2} = 2.75$. The midpoint rule uses the height at the centre of each strip and gives $0.5\,(0.0625 + 0.5625 + 1.5625 + 3.0625) = 2.625$. Both are much closer to the exact $2.667$ than either one-sided sum.

Simpson's rule, which fits parabolas through groups of three points, is exact for this function and for any cubic. Software uses refinements of these ideas to compute integrals to many decimal places in a fraction of a second.

A short history

Finding areas by adding thin slices is ancient. Archimedes found the area of a parabolic segment more than two thousand years ago using a method close to a limit. The breakthrough of the seventeenth century, credited to Newton and Leibniz, was the Fundamental Theorem, which showed that areas could be found by reversing differentiation. Bernhard Riemann gave the sums their modern precise definition in the nineteenth century, which is why they carry his name.

Frequently asked questions