Computing Definite Integrals

The process of finding an integral is known as integration.  Definite integrals are utilized when the limits are defined to generate a specific value.  On the other hand, indefinite integrals are utilized when the limits of the integrant are not conditioned. If the lower limit and the upper limit of the independent variable of a function are specified, the integration is said to be definite integrals. This article will discuss definite integrals and how to compute definite integrals and their properties.

What are Definite Integrals?

The definite integrals are generally used to calculate the volume and area enclosed by the curves. The area required to calculate is mostly used for distinct formulas of the square, rectangle, circles, etc. Many shapes are not simple. Therefore for calculating the volume or the area of that shapes, they need the use of definite integrals. 

Suppose a function f(x) is being defined on an interval [a,b], the definite integration for its limits can be viewed as;

∫ab f(x) = F(x)dx

Where b is the upper limit, and a is the lower limit of the function.

Calculus Fundamental Theorem

The ordinates x = a and x = b are bounded by the area of the region curve f(x) is represented as  ∫ab f(x) = F(x)dx. Suppose the point x is in any random point between the limit a and b, then  ∫ax f(x) = F(x)dx depicts the region bounded between a to x.

By the above-mentioned fundamental theorem of Calculus, it derives two basic fundamental theorems ;

First Calculus Fundamental Theorem

Let F(x) be the continuous function on a closed interval [a, b] and A(x) be the area of function. Then,

A’(x) = F(x), for all x belongs to [a, b].

Second Calculus Fundamental Theorem

Let F(x) be the continuous function on a closed interval [a, b] and let f(x) be the antiderivative

of the functions F(x). Then, 

 ∫ab f(x) = F(x)dx = [f(x)]ab = f(b) – f(a)

Area Under Curve

The area integrated under the curve is given by definite integration.

Generally, in cases, the area comes out to be positive only, but in some cases of areas under a curve, the value comes out to be negative.

For this type, the area can be represented as;

 ∫ab f(x) = F(x)dx

Integrals of Rational Function in Definite Integrals

For determining the definite integral of this type of function, the functions are broken down into algebraic use.

Let us understand this with the help of an example ;

1.  Determine the following integral

 ∫13 1-x/x2dx

Solution:

f(x) =  ∫13 1-x/x2dx

=   ∫13 (1/x2 – 1/x) dx

 = ∫13(1/x2)dx –   ∫13(1/x)dx

= 1 – ⅓ – (log(3) – log(1))

= ⅔ – log(3)

2. Determine the following integral 

  ∫01 (x2+x)/x dx

Solution

f(x) =  ∫01 (x2+x)/x dx

=  ∫01 xdx +  ∫01 1dx

= [x2/2]01 + [x]10

= ½ +1

= 3/2 

3.  Determine the following integral

Solution

 ∫01((x)½ + x)dx

= [x3/2/3/2 + x2]01

= ⅔ + 1

= 5/3

Limit of a Sum of Definite Integral

The Definite integral of a function can be viewed either as a sum limit or if there is any antiderivative (f) for an interval having [a, b]. The definite integral of that function is the difference between that values at a point a and b, respectively. 

Properties of Definite Integrals 

Below are mentioned some of the properties of Definite Integrals;

 ∫ab f(x) dx =  ∫ab  f(t) d(t)

 ∫ab -f(x) dx = –  ∫ba f(x) dx

 ∫aa f(x) dx =  0

 ∫ab f(x) dx =  ∫ac f(x) dx +  ∫cb f(x) dx

Integrals by Parts

Below are mentioned formulas of Definite integral by splitting into by parts;

• ∫02a f (x) dx = ∫0a f (x) dx +∫0a f (2a – x) dx

• ∫02a f (x) dx = 2 ∫0a f (x) dx ………….. if f(2a – x) = f (x).

• ∫02a f (x) dx = 0 … if f (2a – x) = – f(x)

• ∫-aa f(x) dx = 2  ∫0a  f(x) dx … if f(- x) = f(x) or it is an even function

• ∫-aa f(x) dx = 0 … if f(- x) = – f(x) or it is an odd function

Conclusion

Integrals play a very vital role in calculus. Integrals allow us to determine the integrity that gives the derivative of a function and gives functions’ output. Applications of integrals consist of the volume enclosed by the surface, the area under the curve, etc. On the other hand, the previous application mostly refers to indefinite integrals. The procedure of integrals is known as integration. If the upper limit and lower limit of independent variable function are known or specified, the integration is known to be a definite integral. Integral is used to determine volume, area, displacement, and other notations that arise when linking precise data.