**Let f(x) be a function defined in a closed interval [a , b].**

**Then the definite integral represents the algebraic sum of the areas of the region bounded by the curve y = f(x) and the x-axis between the lines x = a and x = b**

**All the regions lying above the x-axis have ‘ positive ‘ areas whereas those lying below the x-axis have ‘ negative ‘ areas.**

__Trapizodial Rule__

__Trapizodial Rule__**For finding the definite integral of a linear portion of a curve we can use the formula of area of trapizium and the rule applied is called trapizodial rule.**

**In the adjoining figure AD represent a linear curve and x = a and x = b are the limits of definite integral than**

**= 1/2)(sum of parallel sides) × height**

**= (1/2)[f(a) + f(b)].(b – a)**

**Example : Evaluate:**

**Solution:**

**Here, f(x) = 2x + 3**

**f(6) = 15 and f(2) = 7**

**= 2 × 22 = 44.**

**First Fundamental Theorem Of Calculus:**

**First Fundamental Theorem Of Calculus:**

**If f (x) is a continuous function on [a, b], then is differentiable at every point x in [a, b] and ∀ x ∈ (a, b) .This is called the first fundamental theorem of calculus.**

__Existence of Anti – derivative of Continuous Functions:__

__Existence of Anti – derivative of Continuous Functions:__**If y = f (x) is continuous on [a, b], then there exists a function F (x) whose derivative on [a, b] is f i.e. every continuous function is the derivative of some of the functions.**

**In other words, every continuous function has an anti derivative. However, not every anti derivative, even when it exists, is expressible in closed form, in terms of elementary functions e.g.**

** , , , **

**In all such cases, the anti derivative is obviously some new function which does not reduce to a combination of a finite number of elementary functions.**

__Second Fundamental Theorem Of Calculus :__

__Second Fundamental Theorem Of Calculus :__**If f(x) is a continuous function on [a, b] and F(x) is any anti derivative of f(x) on [a, b]**

**i.e. F'(x) = f (x) ∀ x ∈ (a, b) , then**

**(also called the Newton-Leibnitz formula).**

**The function F(x) is the integral of f(x) and a and b are the lower and the upper limits of integration.**

**Proof :From the first fundamental theorem**

** (As F'(x) = f(x) given)**

**i.e., the expression within the bracket must be constant in the interval and hence we can write**

** ∀ x ∈ [a , b] , where c is some real constant.**

**Thus **

**and , **

**Hence, **

**The second fundamental theorem is used to calculate the value of the definite integral. A note of caution to you is that f (t) must be continuous in [a, b] or else you will have to partition it into subintervals such that f(x) is continuous in each of the subintervals.**

__Change Of Variables In Definite Integral :__

__Change Of Variables In Definite Integral :__**If the functions f(x) is continuous on [a, b] and the function x = g(t) is continuously differentiable on the interval [t _{1}, t_{2}] and a = g (t1) and b = g (t_{2}), then**

__Improper integrals :__

__Improper integrals :__**Definite integral for which a or b or both are infinite are called improper integrals. These are evaluated in the following manner:**

**(i) **

**(ii) **

**(iii) **

**Another type of improper integral is that in which the integrand is not defined for a point c ∈ [a, b].**

**Illustration : Prove that :**

**Solution: Let **

**Put **

**When x → 0, t → 1 and**

**when x → ∞ , t → ∞**

**Hence**