# Definite Integrals

In Geometry, we know how to calculate the area of a triangular plot of land.

We can also find the area of the plot, if it is in the shape of a rectangle or a Square.

**All the shapes mentioned above** (triangle, rectangle, square)** are regular shapes**, whose area can be found out by the known results called formulas.

Now, if we are **given a shape which is not regular or standard one**, then how do we find its area? We try to approximate its area by associating** it with regular shapes.**

**For finding the area of such irregular shapes, area enclosed between two geometrical shapes** we make use of **Definite Integrals.**

Here, we introduce the **concept of Definite Integrals.**

**A definite integral is denoted by***f(x)dx***where***a*is called the lower limit of the integral &*b*is called the upper limit of the integral.

*f(x)dx***denotes the area of the region bounded by the curve***y*=*f*(*x*), the ordinates*x*=*a*,*x*=*b*and the*x*-axis.

**Since the definite integral denotes the area enclosed, hence its value is always definite or fixed. So, it is called definite integral.**

**Relation between Indefinite & definite integral.**

**If f(x) be a continuous function defined on the closed interval [a, b] & F(x) is its antiderivative i.e. **

*f(x)dx = F(x)*

**, then**

*f(x)dx = = F(b) – F(a)*

The above statement is called** the Second fundamental Theorem of integral calculus.**

**Note**: While evaluating a definite integral the * Constant of Integration*, C disappears at the end and hence its effect is nullified. So, we do not write the constant of integration while evaluating a definite integral.

*Examples*

Now let’s consider some examples on definite integrals .

** Example 1:** Evaluate the definite integral:

*dx*

Let *F(x) = dx = + x = +x*

*dx = = F(2) – F(0)*

= – = – 0 =

**Note: **The constant of Integration is not written here, as it does not change the final value of the definite integral.

** Example 2: **Evaluate:

*dx*

*Let F(x) = dx = + = + *

*dx = = F(1) – F(-1)*

= – = e + – –

*e – = e – *

** Example 3:** Evaluate:

*dx*

*Let F(x) = dx = dx*

=* dx – dx = – *log |*x| = – *log|x|

* dx = = F(5) – F(1)*

= – = –

= – log|5| + log|1| = – log|5| + 0 = – log|5|

** Example 4:** Evaluate:

*dx*

*Let F(x) = dx = dx *

= *dx = 3 dx + 4 CosecxCotx*

= *-3Cotx – 4Cosecx*

* dx = = F – F*

–

= (-3(0)-4(1)) –

= (0 – 4) – = 4 + 3 +4 = -1 +4

** Example 5:** Evaluate

*dx*

*Let F(x) =dx*

= 3* dx – 2 xdx + 7 dx = 3 – 2 + 7 +C*

= – + = – + *x*

*dx = = F(4) – F(1)*

= –

= – = 48 + 98 = =

#### Check Point

Evaluate the following definite integrals:

*(x + 1)dx*-
*dx* *dx**(x – 1)(x – 2)dx**dx*

##### Answer Key

*(x + 1)dx =*-
*dx =* *dx = 7e +19**(x – 1)(x – 2)dx =**dx =*

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