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Integrals
Introduction
Integration as an Inverse Process of Differentiation
Practice: Antiderivatives by Inspection
Methods of Integration
Practice: Integration by Substitution
Practice: Trigonometric Integrals
Integrals of Some Particular Functions
Practice: Standard Forms and Completing the Square
Integration by Partial Fractions
Practice: Partial Fractions
Integration by Parts
Practice: Integration by Parts
Practice: Integrals of Square-Root Quadratics
Definite Integral
Fundamental Theorem of Calculus
Practice: Evaluating Definite Integrals
Evaluation of Definite Integrals by Substitution
Practice: Definite Integrals via Substitution
Some Properties of Definite Integrals
Practice: Properties of Definite Integrals
Miscellaneous Examples
Chapter Review: Mixed Practice
Summary
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Glossary

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IntegralsIntroduction

Reading time: ~10 min

Riya is on the first metro train of the morning. As it pulls away from the platform, the display above the door shows just one number: the train's speed. After 1 second it reads 2 m/s, after 2 seconds 4 m/s, after 3 seconds 6 m/s. After t seconds the speed is v = 2t m/s.

Riya wants a number the display never shows: how far the train has gone. Can a record of speeds be turned into a distance?

At a steady speed, distance is speed × time, which is the area of a rectangle on a speed–time graph. Here the speed keeps changing, but the idea survives: the distance covered is the area under the speed graph, the region between the blue speed line and the time axis.

Drag the clock and stop it at each distance on the list.

Time since the train left: ${t} s

Speed v = 2t m/s against time t s. The purple area is the distance covered so far.

You stopped at 16 m after 4 s, 36 m after 6 s and 64 m after 8 s. So after t seconds the train has covered metres. The purple region is a triangle with base t and height 2t, and \frac{1}{2} \times t \times 2t = t^2.

Now differentiate that distance: \frac{d}{dt}\left(t^2\right) = , which is exactly the speed.

So the speed is the derivative of the distance, and finding the distance from the speed means running a derivative backwards. Notice too that the distance was an area. Those two ideas, undoing a derivative and measuring an area, are the two halves of this chapter.

Differential calculus began with tangents: given a function, how steep is its graph at a point? Integral calculus asks the reverse questions:

(a) Given a derivative, find the function. A function whose derivative is f is called an anti derivative (or primitive) of f. The formula that collects all of them is the indefinite integral of f, and finding it is called integration. Speed to distance, flow rate to water collected, marginal cost to total cost: each is this problem.

(b) Find an area. The area of the region between the graph of a function, the x-axis and two vertical lines leads to the definite integral.

At first the two look unrelated. The metro has already hinted that they are one and the same, and the Fundamental Theorem of Calculus makes that precise. It is the reason definite integrals are a working tool in physics, engineering, economics and probability. Isaac Newton and Gottfried Leibniz discovered it independently in the 1660s and 1670s, and the integral sign \int we still use is Leibniz's.

In this chapter we study indefinite and definite integrals, their basic properties, and the main techniques for finding them.

Sina