JEE Mathematics Area Notes Set 01

Area Under a Curve

For a continuous curve y = f(x) lying above the x-axis, the area bounded by the curve, the x-axis and the ordinates x = a and x = b is \( \int_a^b y\,dx=\int_a^bf(x)dx \). If part of the curve lies below the axis, the area of that part is the negative of the integral, so split at the points where the curve crosses the axis and add the absolute values. Similarly, the area between the curve x = g(y), the y-axis and the lines y = c and y = d is \( \int_c^dx\,dy \).

Area Between Two Curves

If \( f(x)\ge g(x) \) on [a, b], the area between them is \( \int_a^b\{f(x)-g(x)\}dx \). Find the meeting points first, which give the limits. If the curves cross inside the interval, split at the crossing points, always subtracting the lower curve from the upper curve. When the region is easier to slice horizontally (for example regions bounded by parabolas of the form \( x=ay^2 \)), integrate with respect to y.

Standard Areas to Remember

Area bounded by \( y^2=4ax \) and its latus rectum x = a: \( \tfrac83a^2 \).
Area between \( y^2=4ax \) and \( x^2=4by \): \( \dfrac{16}{3}ab \).
Area of the circle \( x^2+y^2=a^2 \): \( \pi a^2 \). Area of the ellipse \( x^2/a^2+y^2/b^2=1 \): \( \pi ab \).
Area enclosed by \( \sqrt x+\sqrt y=\sqrt a \) and the axes: \( a^2/6 \).
Area enclosed by |x| + |y| = a: \( 2a^2 \). Area under one arch of sin x: 2.

Using Symmetry and Modulus

Use symmetry about an axis to double the area of one half. For curves with modulus, draw the graph piece by piece; the area of a region between y = |x - 1| and y = 3 - |x| is found by adding triangular or trapezium pieces. Where the answer depends on a parameter, form the area as a function and set it equal to the given value, as in the problem of the line x = a dividing a region into equal parts.

Area Function

If the area under y = f(x) from 0 to a equals A(a), then by the fundamental theorem \( A'(a)=f(a) \). So when the area is given as an expression in a, differentiate it to recover f. Also for exponentials, the area under \( ce^x \) between p and q is proportional to the change f(q) - f(p).