JEE Mathematics Area Under The Curve Notes Set 01

Area Under a Curve

The area between \(y=f(x)\), the x-axis and the ordinates \(x=a\), \(x=b\) is \(\int_a^b|f(x)|\,dx\). If the curve crosses the x-axis between a and b, split the interval at the crossing points and add the absolute values of the pieces. Areas along the y-axis use \(\int_c^d|g(y)|\,dy\) for \(x=g(y)\).

Always sketch first. For even or odd functions use symmetry: for an odd curve such as \(y=x^3-4x\) the two loops have equal areas, and for a curve symmetric about the x-axis (such as \(y^2=4x\)) double the area above the axis.

Area Between Two Curves

If \(f(x)\ge g(x)\) on \([a,b]\), the area is \(\int_a^b[f(x)-g(x)]\,dx\). Find the limits as the x-coordinates of the intersection points. If the curves cross inside the interval, split there. When horizontal strips are simpler, use \(\int[x_R-x_L]\,dy\).

Standard Results

Area of a parabola segment between \(y=ax^2+bx+c\) and the x-axis with roots \(\alpha<\beta\): \(\frac{|a|(\beta-\alpha)^3}{6}\). Area of the circle \(x^2+y^2=a^2\): \(\pi a^2\); of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\): \(\pi ab\). Area of one arch of \(y=\sin x\): 2. The area between \(y^2=4ax\) and \(x^2=4by\) is \(\frac{16ab}{3}\). Area of a segment of a circle subtending angle \(\theta\) at the centre: \(\frac{a^2}{2}(\theta-\sin\theta)\).

Worked Examples

Example 1. The area bounded by \(y^2=x\) and \(2y=x\): the curves meet at \(y=0\) and \(y=2\), so the area is \(\int_0^2(2y-y^2)\,dy=\frac43\).

Example 2. The area between \(y=\ln x\) and \(y=(\ln x)^2\): they meet at \(x=1\) and \(x=e\), and the area is \(\int_1^e(\ln x-\ln^2x)\,dx=3-e\).

Example 3. The smaller part of the circle \(x^2+y^2=a^2\) cut by \(x=\frac a{\sqrt2}\) subtends a right angle at the centre, so its area is \(\frac{a^2}{2}\left(\frac\pi2-1\right)\).

Quick Tips

Draw the region, mark the intersection points, and decide whether vertical or horizontal strips give a single integral. Greatest integer functions give rectangles; absolute values give symmetric pieces.