Chapter 8: Application of Integrals

The definite integral $\int_a^b f(x)\,dx$ measures the signed area between the graph of $f$ and the $x$ -axis. In this chapter we turn that single idea into a versatile area-computing tool. We will find the area enclosed by a parabola and a line, the region between two intersecting curves, the area inside an ellipse, and a host of other shapes whose boundaries are described by equations rather than by Euclidean rulers and compasses.

The procedure is always the same. Step 1: sketch. Find where the curves meet, identify the bounded region, and note which curve is on top. Step 2: set up. Choose to integrate along $x$ or along $y$ , pick whichever gives cleaner limits. Step 3: integrate. Evaluate using the techniques from Chapter 7. The hardest part is almost always Step 1; if your sketch is wrong, the answer is wrong.

For board exams, expect a single big problem: a region bounded by one or two specific curves, often a parabola together with a line, or a circle and a line, sometimes an ellipse. The answer comes out to a clean number , usually involving $\pi$ or a small rational.

For JEE, the difficulty is amplified by parameters and by the requirement to choose the right slicing direction. Some regions are described much more cleanly by horizontal slices ( $dy$ ) than by vertical slices ( $dx$ ), and vice versa. You should be able to switch fluently between the two.

Prerequisites: the definite-integral techniques from Chapter 7 (substitution, completing the square, even/odd symmetry), and a comfort with coordinate geometry from Class XI (the standard equations of the parabola, ellipse, circle, and lines).

A philosophical note. The fact that areas , sums of infinitesimal rectangles , equal differences of antiderivative values is the Fundamental Theorem. This chapter is, in a sense, the Fundamental Theorem applied to geometry. Each region you compute is a small testimony to the same deep result.

What's inside

Area under a simple curve , the basic setup with $y = f(x)$ above the $x$ -axis.
Area between two curves , top minus bottom, with care at intersections.
Area bounded by a parabola and a line , the classic exam problem.
Area bounded by a circle, ellipse, or other curve , using symmetry and standard integrals.
Choosing $dx$ vs $dy$ , when horizontal slices win.
Composite regions , splitting at intersection points.

Key results / Formula card

Setup	Formula
Area under $y = f(x) \ge 0$ on $[a, b]$	$\int_a^b f(x)\,dx$
Area between $y = f(x)$ and $y = g(x)$ , $f \ge g$ on $[a, b]$	$\int_a^b [f(x) - g(x)]\,dx$
Area between $x = u(y)$ and $x = v(y)$ , $u \ge v$ on $[c, d]$	$\int_c^d [u(y) - v(y)]\,dy$
Circle $x^2 + y^2 = r^2$	$\pi r^2$ (whole area)
Ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$	$\pi a b$
Parabola $y^2 = 4ax$ in $[0, h]$	$\dfrac{4}{3}\sqrt{a h^3}$ between curve and chord at $x = h$

How to read this chapter

Always sketch first, even if rough. Identify points of intersection by solving the simultaneous equations. Decide whether $dx$ or $dy$ gives cleaner limits , sometimes the same problem becomes trivial with the other slice direction. Use symmetry whenever possible: for a region symmetric about an axis, compute half and double.

Chapter 8: Application of Integrals

What's inside

Key results / Formula card

How to read this chapter

Sub-topics

Practice quiz