Leibniz’s Rule
If f is continuous function on [a, b] and u(x) and v(x) are differentiable functions of x whose values lie in [a, b] then
= f(v(x)) – f(u(x))
Inequalities
 If at every point x of an interval [a, b] the inequalities g(x) ≤ f (x) ≤ h (x) are fulfilled then dx ≤ dx ≤ dx, a < b
 If m is the least value (global minimum) and M is the greatest value (global maximum) of the function f(x) on the interval [a, b] (estimation of an integral). Then m(b – a) ≤ dx ≤ M (b – a)
 ≤
Different cases of bounded area










Curve tracing
To find the approximate shape of a curve, the following procedure is adopted in order:
 Symmetry:
 Symmetry about xaxis: If all the powers of “y” in the equation are even then the curve is symmetrical about the xaxis, e.g., y^{2} = 4ax.
 Symmetry about yaxis: If all the powers of “x” in the equation are even then the curve is symmetrical about the yaxis, x^{2} = 4ay.
 Symmetry about both axis: If all the powers of “x” and “y” in the equation are even, the curve is symmetrical about the axis of “x” as well as “y”, e.g., x^{2} + y^{2} = a^{2}.
 Symmetry about the line y = x: If the equation of the curve remains unchanged on interchanging “x” and “y”, then the curve is symmetrical about the line y = x, e.g., x^{3} + y^{3} = 3xy.
 Find the points where the curve crosses the xaxis and the yaxis.
 Find dy/dx and examine if possible the intervals when f(x) is increasing or decreasing and also stationary points.
 Examine what happens to “y” when x → ∞ or x → –∞.
Some standard area
 Area bounded by y = sin x, 0 ≤ x ≤ π and xaxis is 2 sq. units. In fact area of one loop of y = sin x and y = cos x is 2 sq. units
 Area bounded by y = log_{e}x, y = 0 and x = 0 is 1 sq. unit.
 Area of ellipse is π ab sq. units.
 Area bounded by y^{2} = 4ax and x^{2} = 4by, a > 0; b > 0 is = sq. units.