(628)-272-0788 info@etutorworld.com
Select Page

# 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 = ax = band 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 [ab] & 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 2Evaluate:  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:

1. (x + 1)dx
2.  dx
3. dx
4. (x – 1)(x – 2)dx
5. dx
1. (x + 1)dx =
2.  dx =
3. dx = 7e +19
4. (x – 1)(x – 2)dx =
5. dx =

## Personalized Online Tutoring

eTutorWorld offers affordable one-on-one live tutoring over the web for Grades 2-12, Test Prep help for Standardized tests like SCAT, CogAT, SSAT, SAT, ACT, ISEE and AP. You may schedule online tutoring lessons at your personal scheduled times, all with a Money-Back Guarantee. The first one-on-one online tutoring lesson is always FREE, no purchase obligation, no credit card required.

For answers/solutions to any question or to learn concepts, take a FREE Demo Session.

No credit card required, no obligation to purchase.
Just schedule a FREE Sessions to meet a tutor and get help on any topic you want!

## Pricing for Online Tutoring

Tutoring Package Validity Grade (1-12), College
5 sessions 1 Month $139 1 session 1 Month$28
10 sessions 3 months $269 15 sessions 3 months$399
20 sessions 4 months $499 50 sessions 6 months$1189
100 sessions 12 months \$2249

## IN THE NEWS

Our mission is to provide high quality online tutoring services, using state of the art Internet technology, to school students worldwide.