What is an integral?

What does an integral represent?

Suppose a car's speed keeps changing. If the speed were constant, distance would just be speed × time. With changing speed, you can chop the journey into tiny time slices, treat the speed as constant in each, and add up all the little distances. An integral does exactly that, with slices so thin the answer becomes exact. On a speed against time graph, that total is the area under the curve.

So an integral can stand for any total built from a changing rate: distance from speed, water in a tank from the flow rate, or energy from power. The symbol ∫ is a stretched S, for "sum".

Integral notation

• ∫ is the integral sign.
• a and b are the limits: the lower and upper ends of the interval.
• f(x) is the function being integrated, called the integrand.
• dx says the variable is x, and stands for the width of the tiny slices.

Area under a curve: adding up rectangles

Take y = x² + 1 between x = 0 and x = 3.

Here x is the input and y = x² + 1. To estimate the area under it, split the interval into rectangles of equal width and add their areas. With 3 rectangles of width 1, using the height at each rectangle's right edge (x = 1, 2, 3), the heights are 2, 5 and 10, giving an area of 2 + 5 + 10 = 17. Using the left edges instead (x = 0, 1, 2) gives 1 + 2 + 5 = 8. The true area is between the two. More, thinner rectangles close the gap:

The sums close in on 12. These sums are called Riemann sums, and the definite integral is defined as the limit they approach as the rectangles get infinitely thin, as set out in OpenStax Calculus. Drag the number of rectangles yourself in the Calculus course.

Integration is the reverse of differentiation

Adding up rectangles works, but it is slow. The shortcut is the most important result in calculus, the fundamental theorem of calculus (OpenStax): to find the area, find a function F whose derivative is f, then subtract its values at the limits.

F is called an antiderivative of f. For y = x² + 1, an antiderivative is F(x) = x³/3 + x, because differentiating it gives x² + 1. So the area is F(3) − F(0) = (9 + 3) − 0 = 12, exactly what the rectangles were heading towards. If derivatives are new to you, see our guide to what a derivative is.

Indefinite integrals and + C

An integral without limits is an indefinite integral: the answer is a function, not a number. The reverse of the power rule is to add one to the power and divide by the new power:

The + C is the constant of integration. Constants vanish when you differentiate, so x² + 5 and x² − 3 both have derivative 2x. When you integrate 2x, you can't tell which constant was there, so you write x² + C.

Check by differentiating: the derivative of 2x³ − 2x² + 5x + C is 6x² − 4x + 5, the function we started with. Our Integral Calculator shows the working for harder integrals, and its guide covers more methods: see the integral calculator guide and integration by parts.

Definite integrals: worked examples

Example 1: an area

No + C is needed in a definite integral: it would appear in both F(b) and F(a) and cancel.

Example 2: distance from speed

A drone's upward velocity is v = 3t² metres per second, t seconds after take-off. How far does it rise in the first 4 seconds?

It rises 64 m. Integrating velocity gives displacement, just as differentiating displacement gives velocity. Try one: this question asks you to evaluate a definite integral.

Area below the axis counts as negative

Where a curve is below the x-axis, the integral comes out negative, because the heights f(x) are negative. If a curve dips below and comes back above, the positive and negative parts can cancel. For the total shaded area, split the integral where the curve crosses the axis, work out each piece, and add up their sizes ignoring the signs.

Common mistakes

• Forgetting + C in an indefinite integral.
• Differentiating instead of integrating. Integration raises the power by one; differentiation lowers it.
• Subtracting the wrong way round. It is F(upper limit) − F(lower limit).
• Using the power rule on 1/x. It fails for n = −1; the integral of 1/x is ln|x| + C.
• Treating a negative integral as a negative area. Areas are positive; split the integral where the curve crosses the axis.
• Dropping dx. It tells you which variable you are integrating with respect to.

Read next

Maths · CalculusIntegral calculator with steps: definite and indefiniteFree integral calculator with steps: indefinite integrals, definite integrals with exact values, improper integrals and the area under the curve shaded.