Application of IntegralsIntroduction

Put the field on a graph, with one unit for every 10 metres. The road is the x-axis, the hedges are the lines and , and the canal bank follows the curve . Now use the idea from Chapter 7: cut the field into thin vertical strips. Each strip is very nearly a rectangle. The filled strips fit inside the field and the dashed strips cover it completely, so the true area is trapped between their two totals.

Both totals close in on one number, and that number is the area of the field. In Chapter 7 you met this limit of a sum as a definite integral, and the Fundamental Theorem of Calculus works it out in one line: 26/3828/3.

So the field measures square units. One square unit is 10 m × 10 m = 100 m², so the field is about 867 m² — a number no formula in the patwari's notebook could have given.

Beyond the Formula Book

The formulae of school geometry are powerful: they handle every figure with straight sides, and the circle as well. But as soon as a boundary is some other curve, they have nothing to say. That is where integral calculus takes over.

Which of these regions needs integration to find its area?

In Chapter 7 you found the area bounded by a curve , the x-axis and the ordinates and as the limit of a sum. This chapter puts that idea to work. You will find the areas under simple curves — straight lines, arcs of circles, parabolas and ellipses, all in standard form — and learn what to do when a curve dips below the x-axis.

From Archimedes to Cauchy

The idea behind this chapter is very old. Greek mathematicians measured curved figures by filling them with simpler shapes that crept closer and closer to the curve. This method of exhaustion goes back to Eudoxus, and Archimedes used it to find, among other things, the area of a segment of a parabola — an early form of integration.

The systematic theory came some two thousand years later. In 1665 Isaac Newton began his "theory of fluxions", and treated the antiderivative as the inverse of finding tangents. Between 1684 and 1686 Gottfried Leibniz published a paper he called Calculus summatorius, about adding up infinitely many infinitely small areas, and he wrote that sum with a tall S: . In 1696, on a suggestion from J. Bernoulli, he renamed it calculus integrali. Newton and Leibniz worked independently, by very different routes, and reached practically the same results; Leibniz was the first to see clearly how the antiderivative and the definite integral are tied together.

One question was still open: what exactly is the sum of infinitely many infinitely thin strips? The answer, built on the idea of a limit, came only in the early 19th century, from Augustin-Louis Cauchy.

Drag the milestones into order, earliest at the top.

The tall S that Leibniz chose stands for summa, a sum. Every area in this chapter is exactly that: a sum of thin strips, pushed to the limit.