Definite Integrals

Everything in this world is made up of different shapes. Some are definite, while some are indefinite in manner. Now, let’s look into the shapes to determine the present area. For definite shapes, we can apply formulas to find the area. However, for indefinite shapes, we need to divide the shapes into some numbers to make the shapes definite, but this method cannot be applied to every shape. For such shapes, the area can be determined using the method of integration. The method of definite integrals is also used to determine the given area of curves for which the curve’s equation is known.

What do you mean by definite integrals?

Definite integrals represent the area under the curve between two fixed limits. These integrals help determine the area of a curve in a graph. In evaluating definite integrals, one can take limit points as (a,b) and find the area of the curve concerning the x-axis.  

You can represent the definite integrals as:

∫baf(x)dx

where,

a stands for the lower limit, and 

b denotes the upper limit. 

While integration stands for adding areas, definite integrals are the sum of areas with definite limits. 

The formula of definite integrals:

∫abf(x)dx=F(b)−F(a)

Here, F(a) is the lower limit value of the integral, and F(b) is the upper limit value of the integral. The numbers a and b at the bottom and top of the integral sign are the lower and upper limits. Although you have the value of b and a, the lower limit can be higher than the upper limit. Therefore, b and a are termed as intervals of integrals.  

Definite integrals properties:

The definite integral properties help for finding the integral for a function multiplied by a constant, for the sum of the functions, and for even and odd functions. Given below are the properties:

1st property:  p∫q f(x) dx = p∫q f(y) dy

2nd property: p∫q f(x) d(x) = – q∫p f(x) d(x), also p∫p f(x) d(x) = 0

3rd property:  p∫q f(x) d(x) = p∫r f(x) d(x) + r∫q f(x) d(x)

4th property: p∫q f(x) d(x) = p∫q f( p + q – x) d(x)

5th property: o∫p f(x) d(x) = o∫p f(p – x) d(x)

6th property: ∫02p f(x)dx = ∫0p f(x)dx +∫0p f(2p-x)dx…if f(2p-x) = f(x)

7th property: 

• ∫02 f(x)dx = 2∫0x f(x) dx … if f(2p-x) = f(x)

• ∫02 p f(x)dx = 0 … if f(2p-x) = -f(x)

8th property:

• ∫-pp f(x)dx = 2∫0p f(x) dx … if f(-x) = f(x) or the it can be even functions

• ∫-ppf(x)dx = 0 … if f(2p-x) = -f(x) or they are odd functions

How can you determine definite integrals? 

• The primary step for determining definite integrals is to determine the antiderivative that is F(x)

• The secondary step is to determine the values of F(x) and F(y)

• The last step is to find F(x)-F(y)

Example:

• ∫21 2y dy

• At y=1: ∫2y dy = 12 + C

• At y=2: ∫2y dy = 22 + C

• (22 + C) − (12 + C)

• 22 + C − 12 − C

• 4 − 1 + C − C = 3

• ∫1 2 2y dy = 3

A = 2+4 × 1/2 = 3

Which has an area of 3.

Definite integrals application:

The definite integrals are primarily used to find the areas of plane figures such as circles, parabolas, ellipses. 

Engineering is the main field that involves integration in many fields. The integration concepts explained the formulae of physics. Wherever the cross-section area of the studying material is non-uniform, we use the integration concept. For example, when the area of a cross-section of the material to find the centre of mass is non-uniform, we use definite integrals to find the coordinates of COM from any reference point. 

The velocity and displacement of the bodies in motion will be calculated using definite integrals. The volume of the indefinite curves is found using the definite integrals. Several physical applications of the definite integral are common in engineering and physics.

Through definite integrals, one can determine the body’s mass if and only if the function of its density is known. You can also determine the value of work by integrating the force function. 

Definite integrals can also determine the force exerted on an object submerged in a liquid.

Conclusion:

Many fields involve definite integrals in real life. The integration of the curves was the easiest and most efficient way once we got to use it. Most people say the concepts we learned in high school were not useful to us in real life. But that’s not true; the things we learn in high school are the basics of engineering mathematics.