Integrals

The idea in plain language

An integral adds things up. Suppose you want the area of a region under a curved graph. You can approximate it with shapes whose area you know, namely rectangles, and add their areas together.^[2] The approximations get better and better as the number of rectangles grows.^[2] The integral is the value those sums approach in the limit.^{[3, 10]}

The Calculus page calls this the “area problem”. Its partner, the Derivatives page, is about rates of change. This page shows how integrals are defined and written, how the Fundamental Theorem of Calculus joins the two halves of calculus, and where integrals are used.

How it works: from rectangles to an integral

Riemann sums. Split an interval [a, b] into n pieces of width Δx, pick any point c_i in each piece, and use the height of the curve there, f(c_i), as the height of a rectangle.^[10] The total of these rectangle areas, f(c₁)Δx + f(c₂)Δx + … + f(c_n)Δx, is called a Riemann sum.^{[10, 2]} It is named after the 19th-century mathematician Bernhard Riemann, who developed the idea.^[2]

A small example. OpenStax asks for the area under f(x) = x² between 0 and 2, using four rectangles.^[2] Each rectangle is 0.5 wide. Using the left edge of each piece for the height gives 0.5 × (0 + 0.25 + 1 + 2.25) = 1.75, and using the right edge gives 0.5 × (0.25 + 1 + 2.25 + 4) = 3.75 (our own working).^[2] With more rectangles the right-edge sum shrinks: we get 3.08 with 10 rectangles and about 2.71 with 100 (our own working, using the same right-edge method). It is heading for the exact area, 8/3 or about 2.67, which the Fundamental Theorem gives further down.^{[2, 3, 4, 1]}

The definite integral. OpenStax defines the definite integral of f from a to b as the limit of the Riemann sums as the number of rectangles n goes to infinity.^{[3, 10]} It is written ∫_aᵇ f(x) dx. The function f(x) is the integrand, and dx names x as the variable of integration.^[3] Every continuous function on [a, b] has an integral there.^[3]

Net signed area

For a curve that stays above the x-axis, the definite integral is the area under it.^{[10, 2]} When the curve dips below the axis, the part below counts as negative: OpenStax calls the area above the axis minus the area below it the net signed area.^[3] It can be positive, negative or zero.^[3]

Motion shows why this is useful. If the curve is a velocity, the area under it tells you how far the object is from where it started, which is its displacement.^[3] In OpenStax’s car example, 2 hours north at 60 mph and then 3 hours south at 40 mph put the car back at its starting position: the integral of its velocity is 120 − 120 = 0.^[3] The total distance driven is found by integrating the absolute value of the velocity, which gives 120 + 120 = 240 miles.^[3]

Antiderivatives and the indefinite integral

An antiderivative of f is a function whose derivative is f.^[1] Because 2x is the derivative of x², for instance, x² is an antiderivative of 2x.^[1] But x² plus any constant is one too, and MIT’s notes explain why: if two functions have the same derivative, they differ by a constant, a fact they call the foundation of all of calculus.^[11] So the most general antiderivative is written F(x) + C.^[1]

The symbol ∫ is the integral sign, and ∫f(x) dx, written without limits, is the indefinite integral: the whole family F(x) + C.^[1] The two notations look alike, but an indefinite integral is not the same thing as a definite integral.^[3] A definite integral is a number.^[4]

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus establishes the relationship between differentiation and integration, and gives a way to evaluate definite integrals without Riemann sums.^[4] This page follows OpenStax’s numbering of its two parts.^[4] MIT’s 18.01SC notes number them the other way round, so check which convention a book uses.^[11]

Part 1 (OpenStax). Let f be continuous on [a, b], and define an “area so far” function F(x) = ∫_aˣ f(t) dt. Then F′(x) = f(x).^[4] In words: the rate at which the accumulated area grows is the height of the curve.^[4] MIT’s notes call this result the second fundamental theorem.^[11] It also means that a continuous function always has an antiderivative, namely this area function.^[4]

Part 2 (OpenStax), also called the evaluation theorem. If f is continuous on [a, b] and F is any antiderivative of f, then ∫_aᵇ f(x) dx = F(b) − F(a).^[4] MIT’s notes call this the first fundamental theorem and put it in words: the integral of a derivative is the difference between two outputs of the original function.^[11] The difference F(b) − F(a) is often written F(x)|_aᵇ.^[4] The ”+ C” can be left out here, because it cancels in the subtraction.^[4]

Back to x² on [0, 2]: an antiderivative is x³/3, so the integral is 8/3 − 0 = 8/3, the same value the rectangles were approaching (our own working).^{[4, 1]}

The math (optional)

The definition, as OpenStax writes it, with x_i* an arbitrary point in the i-th piece:^{[2, 3]}

∫_aᵇ f(x) dx = lim (n → ∞) Σ_i=1ⁿ f(x_i*) Δx

Basic rules for indefinite integrals, each from OpenStax:

• Power rule: ∫xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + C, for n ≠ −1.^[1]
• The case n = −1: ∫(1/x) dx = ln|x| + C.^[1]
• Exponential and trigonometric: ∫eˣ dx = eˣ + C, ∫cos x dx = sin x + C and ∫sin x dx = −cos x + C.^[1]
• Sums and constant multiples: ∫(f(x) ± g(x)) dx = F(x) ± G(x) + C, and ∫k·f(x) dx = k·F(x) + C.^[1]
• Substitution: ∫f(g(x))·g′(x) dx = ∫f(u) du = F(g(x)) + C, with u = g(x).^[6] It is called substitution because part of the integrand is replaced by u and part by du.^[6]

Evaluating integrals of products, quotients or compositions is more complicated than for sums and constant multiples.^[1]

A worked example. OpenStax asks for the definite integral of f(x) = x² − 3x over [1, 3].^[5] By the power rule an antiderivative is x³/3 − 3x²/2, and Part 2 gives (9 − 13.5) − (1/3 − 1.5) = −10/3, about −3.33 (our own working).^{[1, 4]} The answer is negative because the curve lies below the x-axis on this interval, so the region between curve and axis has an area of +10/3 (our own working).^{[3, 4]}

Applications

Net change. The new value of a changing quantity equals its starting value plus the integral of its rate of change.^[5] Negative quantities are handled automatically, with no need to split the calculation into more than one integral.^[5] The motorboat in the examples above is a case of this.^[5]

Area between curves. If one graph f lies above another graph g between x = a and x = b, the area of the region between them is ∫_aᵇ [f(x) − g(x)] dx.^[7]

Work. When a force moves an object, it does work on it.^[8] For a constant force, work is force times distance.^[8] A spring’s force is not constant, and OpenStax uses an integral to find the work done on one: by Hooke’s law, the force needed to stretch or compress it is F(x) = kx.^[8]

Mass from density. For a thin rod whose density ρ(x) varies along its length, the mass is the limit of a Riemann sum, which is the integral of ρ(x) from one end to the other.^[8]

Common misconceptions

“An integral is always an area, so it can’t be negative.” Area is always positive, but a definite integral can produce a negative number, because it is a net signed area.^{[4, 3]} The integral of x² − 3x over [1, 3] is −10/3, even though the region has a positive area (our own working).^{[5, 4]}

“Displacement and distance are the same integral.” Integrating velocity gives displacement, which can be zero after a long trip; distance needs the absolute value of the velocity.^[3]

“Definite and indefinite integrals are the same thing.” They look similar but are not: a definite integral is a number, while an indefinite integral is a family of functions.^{[3, 4, 1]}

“There is a product rule for integrals, like the one for derivatives.” Integrals of products, quotients or compositions are more complicated to evaluate than integrals of sums; substitution is one technique, used for compositions.^{[1, 6]}

“Everyone numbers the Fundamental Theorem the same way.” OpenStax and MIT’s 18.01SC notes use opposite numbering for the two parts.^{[4, 11]}

History

Archimedes. OpenStax traces the rectangle idea back to Archimedes, whose method of exhaustion filled an irregular region with smaller and smaller shapes whose areas could be calculated exactly.^[2] MacTutor calls this method the early form of integration.^[12] In his Quadrature of the Parabola, Archimedes found the area of a segment of a parabola cut off by any chord.^[12]

Newton and Leibniz. Most mathematicians and historians agree that Isaac Newton and Gottfried Leibniz developed calculus independently.^[9] According to MacTutor, Newton’s method of fluxions rested on his insight that integration is the inverse of differentiation.^[14] Leibniz first wrote the ∫ f(x)dx notation in a manuscript dated 21 November 1675.^[13] It first appeared in print in 1686, in a paper on the integral calculus in Acta Eruditorum.^[13] The Stanford Encyclopedia of Philosophy says Leibniz’s 1686 essay De Geometria Recondita “may be said to represent” the official birth of the integral calculus.^[17] The integral sign is an elongated S, suggesting a sum.^[3] MIT’s notes remark that the Σ notation for Riemann sums is meant to recall Leibniz’s notation.^[10]

A bitter dispute over who invented calculus followed.^[18] MacTutor reports that the Royal Society’s 1713 committee report, Commercium epistolicum, found for Newton and was written by Newton himself.^{[13, 14]} The Derivatives page tells that story in more detail.

Rigour. The infinitesimals the founders used were controversial from the start.^[17] MacTutor credits Augustin-Louis Cauchy with a rigorous definition of the integral.^[16] Bernhard Riemann clarified the notion further, defining what is now called the Riemann integral and giving the conditions for a function to have one.^[15]

Going further

This page stays with integrals of a single variable. Improper integrals, numerical methods for approximating integrals and multivariable integrals are left for later pages. The free OpenStax Calculus Volume 1 textbook and MIT’s 18.01SC course, listed below, work through all of this with many more examples.