What is an integral? Adding up infinitely many small pieces

An integral is the total you get by adding up a quantity that changes continuously, split into infinitely many small pieces. If a car's speed keeps changing, the integral of the speed over an hour is the distance covered. On a graph, an integral is the area between a curve and the horizontal axis. It is the second half of calculus, and the fundamental theorem of calculus shows that it undoes the first half, the derivative.

In short

• An integral is accumulation: a rate added up over an interval gives a total, and on a graph that total is the area under the curve.
• It is defined by slicing the area into rectangles, adding them, and taking the limit as the slices get infinitely thin.
• The fundamental theorem of calculus turns that infinite sum into one subtraction, using a function whose derivative is the curve.

What does an integral add up?

When something changes at a steady rate, the total is a multiplication. A car at a steady 60 kilometers an hour for 2 hours covers 60 × 2 = 120 kilometers. Draw that as a graph, speed up the side and time along the bottom, and the speed is a flat line at 60. The distance, 120, is the area of the rectangle under it: 60 high and 2 wide.

When the rate changes, the same idea still holds: the total is the area under the curve. But a curved region has no simple formula, so it has to be built from pieces. That is the whole job of an integral. An integral turns a rate into a total by adding up its effect over every instant. Speed adds up to distance, rainfall per hour adds up to total rainfall, the flow of water through a pipe adds up to the volume delivered.

How do you find the area under a curve?

Take the curve y = x² between x = 0 and x = 3 and ask for the area beneath it. The method is to cover the area with rectangles, add up their areas, and then use more and thinner rectangles.

Step 1, with 3 rectangles. Split 0 to 3 into three strips each 1 wide. Make each rectangle as tall as the curve at the strip's right edge: heights 1², 2² and 3², which are 1, 4 and 9. The total is 1 + 4 + 9 = 14. Those rectangles poke above the curve, so 14 is too much.

Step 2, the other way. Use the height at each strip's left edge instead: 0², 1² and 2², so 0 + 1 + 4 = 5. Those rectangles sit below the curve, so 5 is too little. The true area lies between 5 and 14.

Step 3, more strips. Repeat with 6 strips, each 0.5 wide, then 12 strips, each 0.25 wide, and so on:

Rectangles	Width of each	Left edges (too low)	Right edges (too high)
3	1	5.000	14.000
6	0.5	6.875	11.375
12	0.25	7.906	10.156
100	0.03	8.865	9.135
1,000	0.003	8.987	9.014

The low and high estimates squeeze together, and the number they both close in on is 9. That limit is the integral, and the area under y = x² from 0 to 3 is exactly 9. Using the height at the middle of each strip converges even faster: 8.75 with 3 rectangles, 8.98 with 12. Calculators and computers find integrals in essentially this way when no formula is available.

How is an integral written?

The integral of a function f from a to b is written ∫ₐᵇ f(x) dx. Each part has a meaning. The tall S, ∫, was introduced by Gottfried Wilhelm Leibniz in 1675 as a stretched letter S for summa, Latin for sum. The numbers a and b at the bottom and top are the limits of integration, where the area starts and stops. The f(x) is the height of each thin rectangle, and dx stands for its tiny width, so f(x) dx is one rectangle's area. Read in full, the symbol says: add up the areas of the infinitely thin rectangles from a to b.

So the example above is ∫₀³ x² dx = 9. Bernhard Riemann made the definition precise in the 1850s, which is why these rectangle totals are called Riemann sums.

One detail matters. Area below the horizontal axis counts as negative. The integral of a car's velocity, which is negative when it reverses, gives its net change in position, not the total distance driven. Drive 10 kilometers forward and 10 back and the integral of velocity is 0.

How are integrals connected to derivatives?

The derivative turns a total into a rate; the integral turns a rate back into a total. The fundamental theorem of calculus makes that exact: to integrate f from a to b, find any function F whose derivative is f, called an antiderivative, and compute F(b) − F(a).

For y = x², the antiderivative is x³ ÷ 3, because the derivative of x³ is 3x², and dividing by 3 leaves x². So ∫₀³ x² dx = 3³ ÷ 3 − 0³ ÷ 3 = 27 ÷ 3 − 0 = 9, the same 9 the thousand rectangles were approaching, in one line. That is why the two halves of calculus are taught together: one hard problem, an infinite sum, becomes a much easier one, undoing a derivative.

Where are integrals used?

Distance from speed. A car's speedometer reading, added up over a journey, gives the distance on its odometer. Navigation systems that lose a satellite signal in a tunnel estimate position this way, integrating speed and direction.

Total rainfall. A rain gauge that records the rate of rainfall can give the storm total. If rain falls at 2 millimeters an hour for 3 hours and then 6 millimeters an hour for 1 hour, the area under the rate graph is 2 × 3 + 6 × 1 = 12 millimeters. Real rates change smoothly, and the integral handles that.

Probability. For a quantity that can take any real number in a range, like a person's height, probabilities are areas under a curve whose total area is 1. The chance of a value between two numbers is the integral of the curve between them. For the bell-shaped normal distribution, about 68% of the area lies within one standard deviation of the mean.

Work and energy. Lifting a box against a steady force, the work done is force times distance. When the force changes, as when stretching a spring that pulls back harder the further it stretches, the work is the integral of force over distance. For a spring that is ½kx², where k measures its stiffness and x the stretch, which is the energy stored in it.

Areas and volumes. Integrating the circumference of circles of growing radius, 2πr, from 0 to r gives the area of a circle, πr², the formula behind pi's most familiar use. Archimedes had found areas and volumes by a similar slicing more than 2,000 years earlier.

What do people get wrong about integrals?

The most common misconception is that every integral can be worked out with a formula. Many can't. The bell curve's shape, e^(−x²), has no antiderivative that can be written with ordinary functions, so its areas are computed numerically, with sums like the rectangles above, and printed in tables. The rectangle method is not just a teaching device; it is often the only way.

The second is to think of an integral only as area. Area is the picture; accumulation is the idea. Whenever a quantity builds up from a rate, whether money from a flow of payments, a drug dose from an infusion, or charge in a battery from a current, an integral is the tool that adds it up.

Check yourself

• Without looking back, explain why left-edge rectangles underestimate the area under y = x² and right-edge rectangles overestimate it.
• What does each part of ∫₀³ x² dx stand for?
• How does the fundamental theorem of calculus get the exact answer 9 without adding up any rectangles?

Sources

• Approximating areas — OpenStax, Calculus Volume 1, Rice University, 2016
• The definite integral — OpenStax, Calculus Volume 1, Rice University, 2016
• The fundamental theorem of calculus — OpenStax, Calculus Volume 1, Rice University, 2016
• The rise of calculus — MacTutor History of Mathematics, University of St Andrews
• Integral — Wikipedia