Introduction

Calculus grew out of two independently-motivated 17th century investigations — Sir Isaac Newton (1643–1727) approached it through physics: he built an expression for the area under a curve by considering a momentary increase at a point, so that the fundamental theorem of calculus was, in effect, already built into his calculations — and his own work reached well beyond mathematics, into optics and gravitation. Gottfried Wilhelm Leibnitz (1646–1716), a German philosopher, mathematician and political adviser, developed the differential and integral calculus as a distinguished independent inventor, and it is largely Leibnitz's notation that survives in use today. Both men are credited as independent discoverers of calculus in the mid-17th century.

Calculus is built around two geometric problems that look completely different on the surface but turn out to be inverse operations of each other:

1. The slope problem. Finding the slope of the tangent line to a curve at a point — solved by the limiting process called differentiation (studied in earlier chapters).
2. The area problem. Finding the area of a region bounded by a curve — solved by another limiting process called Integration, the subject of this chapter.

Having studied differential calculus in the two preceding chapters, this chapter develops the fundamentals of integration — starting with indefinite integration, the process of recovering a function from its derivative.

Why integral calculus matters — three illustrative situations from the book:

Situation 1 (the Brachistochrone problem). The shortest distance path between two points $A$ and $B$ (a straight line) is not the same as the shortest time path for a particle sliding under gravity from $A$ to $B$. A curve that dips more steeply near $A$ lets the particle build speed early, so even though the curved path is longer, a considerable portion of it is covered at greater speed, and the total time is less than along the straight line. Finding that optimal curve is a genuine calculus-of-variations question, and it is Integral Calculus that supplies the tool to solve it — this is historically called the Brachistochrone problem.

Situation 2 (measuring irregular shapes). Elementary geometry gives ready-made formulas for the perimeter of a rectangle ($2(l+b)$), the area of a triangle ($\tfrac12 bh$), the curved surface area of a cone ($\pi r l$), and the volume of a sphere ($\tfrac43\pi r^3$) — all regular shapes. But how would you find the length of a wavy curve traced by a function $f$, the area enclosed between two intersecting curves $f$ and $g$, the surface area of a shape whose profile is a curve $f$ revolved about an axis, or the volume of a solid whose outline is likewise curved? These are exactly the questions integral calculus answers, painlessly, once the machinery is in place.

Situation 3 (a real rate problem). A student riding a bike at $24\text{ m/s}$ sees a barrier $40$ metres ahead and brakes at a retardation of $8\text{ m/s}^2$. Will the bike stop before the collision? Answering this needs the reverse of differentiation: given the acceleration (a derivative), recover the velocity, then the position — exactly the chain of antiderivatives this chapter builds towards.

Other natural questions that integration alone can answer: the speed needed to fire a satellite so it never returns to Earth; the radius of the smallest circular disk that can cover every isosceles triangle of a given perimeter; the volume of material removed when a hole is drilled through the centre of a solid sphere; and how much a bacterial population — growing at a rate proportional to the amount present — increases over a given time once its doubling time is known.