Summary Definite Integrals

 

First Fundamental Theorem of Integral Calculus

If f(x) is a continuous function on the closed interval [a, b], and if

Area function is defined by

Second Fundamental Theorem of Integral Calculus

If f (x) is a function continuous on [a, b] then

Evaluation of definite integral by changing limits after suitable substitution.

Step I : Let z = g(x) be the desired substitution, dz = g^' (x) dx

Step II : when x = a, z = g(a)

x = b, z = g(b)

Properties of definite integrals

The area bounded by the curve y = f(x), x-axis, and the ordinates at

The area bounded by the curve x = f(y), y - axis and the abscissas

If f(x) is continuous in [a,b] and crosses the x-axis at x = c in (a, b) then the area bounded by the curve, x - axis and x = a and x = b is

Area between y = f(x) and y = g(x),