Chapter 8 ยท Introduction

Chapter 8 โ€” Application of Integrals

Slice a region into thin strips, add up their areas, and let the strips become infinitely thin: the definite integral of Chapter 7 turns into a machine for measuring areas bounded by lines, parabolas, circles and ellipses.

A farmer in Punjab owns a field with three straight sides โ€” a road and two boundary walls โ€” and a fourth side that follows the curving bank of a canal. The land record needs its area in square metres. For the rectangular part that is one multiplication; for the curved side no school formula applies. Surveyors have always solved this the same way: they cut the field into narrow strips at right angles to the road, measure the length of each strip, multiply by its width and add. The thinner the strips, the better the answer. More than two thousand years ago Archimedes used the same slicing idea to prove that the region cut off from a parabola by a chord is exactly $\tfrac23$ of the rectangle around it โ€” long before anyone had heard of an integral.

In Chapter 7 โ€” Integrals you learnt to evaluate $\displaystyle\int_a^b f(x)\,dx$ as $F(b)-F(a)$. This chapter puts that skill to work. A thin strip of height $y$ and width $dx$ has area $dA=y\,dx$; adding all the strips from $x=a$ to $x=b$ is exactly what the definite integral does. You will find areas under lines and parabolas, use horizontal strips when the curve is easier to write as $x=g(y)$, learn why a curve below the $x$-axis gives a negative integral but a positive area, and finally prove the two most famous area formulas of all: $\pi a^2$ for a circle and $\pi ab$ for an ellipse.

Definite integrals (Ch 7)โ†’Elementary strip $dA=y\,dx$โ†’Area under a curveโ†’Horizontal strips $x\,dy$โ†’Curves below the axisโ†’Parabolas, circles, ellipses
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • explain why the area of a region under a curve is $\displaystyle\int_a^b y\,dx$ by adding up elementary vertical strips;
  • find the area bounded by a curve, the $x$-axis and two vertical lines โ€” and, with horizontal strips, the area bounded by a curve, the $y$-axis and two horizontal lines;
  • handle curves that lie below the $x$-axis or cross it, by splitting the interval at the crossing points and taking the modulus of each piece;
  • find areas of regions bounded by straight lines, parabolas such as $y^2=4ax$ and $x^2=4ay$, circles $x^2+y^2=a^2$ and ellipses $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ and an axis or a vertical line, using symmetry to save work;
  • prove that a circle of radius $a$ has area $\pi a^2$ and an ellipse with semi-axes $a$ and $b$ has area $\pi ab$;
  • answer the Board's 3- and 5-mark area questions and the JEE Main / JEE Advanced problems built on them.
Teaser โ€” How Big Is the Pond?

A garden pond has the shape of the ellipse $\dfrac{x^2}{16}+\dfrac{y^2}{9}=1$ (lengths in metres). Cut it into $n$ vertical strips of equal width, treat each strip as a rectangle whose height is the width of the pond at the strip's midpoint, and add the areas. Play (or drag the slider) to make the strips thinner and watch the total settle on one number. Can you recognise it?

strips $n$โ€”
strip width $\Delta x$ (m)โ€”
sum of strip areas (mยฒ)โ€”
exact area $12\pi$ (mยฒ)โ€”
errorโ€”

Sections in this chapter

8.1

Area under a Curve

Elementary vertical and horizontal strips; area bounded by a curve, an axis and two lines; curves below the $x$-axis and the modulus rule.

$A=\displaystyle\int_a^b y\,dx=\int_c^d x\,dy$
Open section โ†’8.2

Areas Bounded by Lines, Parabolas, Circles and Ellipses

Triangles from integrals; regions of $y^2=4ax$ cut off by a vertical line; segments of circles; the area $\pi ab$ of an ellipse; using symmetry.

$\pi a^2,\ \ \pi ab,\ \ \tfrac83a^2$
Open section โ†’
โœ… Before you start
This chapter uses the definite integral and its evaluation $\displaystyle\int_a^b f(x)\,dx=F(b)-F(a)$ from Chapter 7 โ€” Integrals (Section 7.4), together with the standard integrals of $x^n$, $\sin x$, $\cos x$ and, for circles and ellipses, the integral $\displaystyle\int\sqrt{a^2-x^2}\,dx$ $=\dfrac x2\sqrt{a^2-x^2}+\dfrac{a^2}2\sin^{-1}\dfrac xa+C$ (Section 7.3). You should also be able to sketch straight lines (Class 11 Mathematics, Ch. 9 โ€” Straight Lines) and the standard parabola, circle and ellipse (Class 11 Mathematics, Ch. 10 โ€” Conic Sections), and recall the principal values of $\sin^{-1}$ (Section 2.1). Quick self-test: evaluate $\displaystyle\int_1^4\sqrt x\,dx$, and give the vertex, focus and latus rectum of $y^2=8x$. (Answers: $\tfrac{14}{3}$; vertex $(0,0)$, focus $(2,0)$, latus rectum the segment of $x=2$ from $(2,-4)$ to $(2,4)$, length $8$.)
โ† 7.4 Definite Integrals, the Fundamental Theorem and Their Properties8.1 Area under a Curve โ†’