What Is an Integral in Calculus? Area, Sums, Examples

A speedometer reports how fast you are going, never how far you have gone. Add up the speedometer readings over a trip and you recover the distance: the total change in position is what the readings add up to⁷. That adding-up of a rate that keeps changing is the job of the integral.

What is an integral in simple terms?

An integral is the continuous analog of a sum, and it is used to calculate areas, volumes and their generalizations⁶. An ordinary sum adds a list of separate numbers. An integral adds a quantity that varies smoothly, so there is no list, only a curve.

Integration is one of the two fundamental operations of calculus, along with differentiation⁶. Differentiation takes a quantity and gives its rate of change. Integration goes the other way and rebuilds the total from the rate. Many quantities in physics, quantities of work among them, may be interpreted as the area under a curve⁸.

A derivative zooms in on one instant. An integral adds up all of them.

Integrals as area under a curve

The picture to hold is area. Take a function that stays above the x-axis. The definite integral from a to b is the area bounded by the graph above, the x-axis below, and the vertical lines at a and b¹⁶.

The curve makes the shape irregular, so a formula for the area is not obvious. The classic way around it is to fill the region with shapes of known area. Archimedes studied the areas of shapes with a process now called the method of exhaustion, which used smaller and smaller shapes to fill an irregular region and obtain closer and closer approximations to the total area¹. The first documented systematic technique for determining integrals is the method of exhaustion of the ancient Greek mathematicians Democritus and Eudoxus in the 4th century BC⁶.

The modern version uses rectangles. By dividing a region into many small shapes that have known area formulas, we can sum these areas and obtain a reasonable estimate of the true area¹. Use smaller and smaller rectangles and the approximations get closer and closer to the area¹. Taking a limit then gives the exact area under the curve¹. That limit of rectangle sums is the definition of the definite integral, and the diagram at the top of this page shows 4 of those rectangles.

What to know before integration

Integration leans on 3 earlier ideas. Limits come first, because taking a limit is what turns an estimate from rectangles into the exact area¹. Derivatives come second, because the Fundamental Theorem works by reversing them³. Sigma (summation) notation comes third: it writes the long sums of rectangle areas compactly¹. If any of these is shaky, the pages on limits and derivatives cover them.

Worked example (our arithmetic): the area under y = x² from 0 to 2

The function and the method follow the textbook problem on estimating an area with rectangles⁴. In each interval we form a rectangle whose height is the function value at a specific point in the interval⁴. Here the interval from 0 to 2 is cut into 4 pieces of width 0.5, and then into 8 pieces of width 0.25:

For this increasing function the right edges overshoot: choosing the height at the right end makes each rectangle take in more area than the graph⁴. The left edges undershoot, because each rectangle misses some of the area⁴. The midpoint estimate does quite well⁴. For a decreasing function the pattern reverses: right endpoints underestimate and left endpoints overestimate⁴. Doubling the rectangles from 4 to 8 pulls the right-edge estimate from 3.75 down to 3.1875, closer to the true value.

The exact answer comes from the Fundamental Theorem of Calculus, covered below. An antiderivative of x² is x³/3, so the area is the value at 2 minus the value at 0³⁷:

The estimates 3.75, 3.1875 and 2.625 all sit near 2.6667, and refining the rectangles closes the gap¹⁴.

Definite integral vs indefinite integral

The word “integral” covers related things, and the difference matters. A definite integral is a number². An indefinite integral asks a different question: what function did we differentiate to get this function⁵. The answer is not a single function. There are infinitely many functions that would work, and they all differ by a constant⁵.

Often people say integral and let the context decide which one is meant⁵. Thinking of the indefinite integral as “what did we differentiate” is the habit to keep⁵. For example, x² has derivative 2x, so an antiderivative of 2x is x², and so is x² + 5 and x² − 9, a different constant each time.

Signed area: when the curve dips below the axis

The first definition of area assumed the graph stays above the axis, and real problems do not always meet that restriction². The definite integral generalizes the concept of the area under a curve for exactly that reason². Areas above the horizontal axis count as positive and areas below count as negative⁶. The integral computes the signed area⁶.

So net signed area can be positive, negative or zero². If the area above the x-axis is larger, the net signed area is positive; if the area below is larger, it is negative².

Worked example (our arithmetic): the function f(x) = 2x between x = −3 and x = 3. The graph is a straight line that forms 2 triangles, 1 above and 1 below the axis, and the geometry is a textbook example²:

The integral is 0 even though the shaded area is not². Zero here means the pieces cancel, not that nothing is there. Keep that in mind whenever a problem asks for total area rather than net change.

The Fundamental Theorem of Calculus

Computing a definite integral from rectangle sums is cumbersome, and so is using only geometric area formulas³. The way out relies on the relationship between differentiation and integration³. That relationship was discovered and explored by Newton and Leibniz during the late 1600s and early 1700s³. The Fundamental Theorem of Calculus is an extremely powerful theorem that establishes the relationship between differentiation and integration, and gives a way to evaluate definite integrals without using Riemann sums or calculating areas³.

If f is continuous and F′ = f, then the integral of f from a to b equals F(b) − F(a)⁷. In words: find any antiderivative, evaluate it at both ends, and subtract. This is why the indefinite integral matters. The major advance in integration came in the 17th century with the independent discovery of the theorem by Leibniz and Newton⁶. The theorem demonstrates a connection between integration and differentiation, and the connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals⁶.

Worked example (our arithmetic): the area under one hump of the sine curve, a standard example in the MIT lecture⁷. An antiderivative of sin x is −cos x:

The area under one hump of sin x is exactly 2⁷. No rectangles, no limit by hand. That is the whole point of the theorem, and the page on the derivative rules is the other half of the toolbox, because every differentiation rule read backward is an integration fact.

Applications of integrals

Integrals appear in many practical situations⁶. Take a swimming pool. From the length, width and depth of a rectangular pool with a flat bottom, you can work out the volume of water it holds. If the pool is oval with a rounded bottom, integrals are required to find exact values for those quantities⁶. In each case, one may divide the sought quantity into infinitely many infinitesimal pieces and sum the pieces to achieve an accurate approximation⁶.

The same move recovers a distance from a speed. The speedometer readings on one side of the theorem add up to the change in position on the other⁷.

Worked example (our arithmetic): a car starts from rest and speeds up so that its speed is v(t) = 4t meters per second, for t from 0 to 5 seconds. The numbers are invented for this example, and the method is the theorem above⁷:

Work in physics is another quantity read as an area under a curve⁸. In every application the question is the same: what is being added up, and over what interval.

A short history of the integral

The early work of Newton and Leibniz lacked rigor. Calculus acquired a firmer footing with the development of limits⁶, which is why the page on limits is worth reading next if the phrase “limit of rectangle sums” felt like a leap.

Integral calculus in one paragraph

Integral calculus is the study of integrals and what they do. It has a computing side, where a definite integral is a number found as a limit of sums¹². It has a reversing side, where an indefinite integral recovers the function that was differentiated⁵. The Fundamental Theorem of Calculus links the 2 and lets a definite integral be evaluated without Riemann sums³. And it has an applying side, where the sum of many small pieces gives volumes, areas and work⁶⁸.

Common mistakes

• Forgetting the constant. The indefinite integral is a family of functions, so a constant of integration belongs in the answer⁵.
• Confusing area with net signed area. The definite integral can be 0, positive or negative, because areas below the axis count as negative²⁶.
• Treating the right-edge estimate as exact. For an increasing function it overestimates, and for a decreasing function it underestimates⁴.
• Mixing up the kinds of integral. A definite integral is a number, and an indefinite integral is a family of functions²⁵.

Where to go next

Integrals sit beside the other core ideas. The limit explains what “closer and closer” means, and the derivative explains the rate that an integral adds up. The rest of the core ideas section builds on all three. For integrals in everyday scenes, see the tub that fills in a normal morning. The open texts used here are free to read: OpenStax on the definite integral, OpenStax on the Fundamental Theorem, Lamar University on the area problem and the MIT lecture.