Revision: 12th Std >> Definite Integration MAH-MHT CET (PCM/PCB) Definite Integration

Definition: Area Function

If a function f is continuous on an interval, the area function is defined by

\[A(x) = \int_{a}^{x} f(t) \, dt\]

This means that A(x) gives the area accumulated from x = a to a variable point x.

Definition: Definite Integral as Limit of Sum

Let ( f(x) ) be a continuous function in the closed interval [a, b] and (h) be the length of (n) equal subintervals, then

\[\int_{a}^{b}f(x)dx=\lim_{n\to\infty}h\sum_{r=0}^{n}f(a+rh)\]

Theorem: First Fundamental Theorem

If f is continuous on [a, b] and

\[A(x) = \int_{a}^{x} f(t) \, dt\]

then A'(x) = f(x) for every x in (a, b).

This means the derivative of the accumulated area

function is the original function itself.

Theorem: Second Fundamental Theorem

If f is continuous on [a, b] and F is any antiderivative of f, then

\[\int_{a}^{b} f(x) \, dx = F(b) - F(a)\]

This is the formula most often used in exams to evaluate definite integrals.

Key Points: Fundamental Theorem of Integral Calculus

The theorem connects differentiation and integration.
If \[A(x) = \int_{a}^{x} f(t) \, dt\], then \[A'(x) = f(x)\].
If F'(x) = f(x), then \[\int_{a}^{b} f(x) \, dx = F(b) - F(a)\].
The result is used to evaluate definite integrals quickly.
The function should be continuous on the interval for direct use of the theorem.

Key Points: Method for Finding Definite Intergrals