Integral Calculus: Evaluation of Definite Integrals

Mathematicians Gottfried Leibniz and Isaac Newton independently invented calculus in the 17th century. 

This branch of mathematics studies continuous or unbroken change; it is a unit that studies the rate of change of matter. Calculus helps understand how matter, stars, and particles change and move in real-time. 

Before the advent of calculus, maths was static. Calculus presented the idea that no object is always at rest, neither the stars in space nor the cells in our body. 

Calculus has applications in economics, medicine, physics, and engineering. Some applications include building safer structures and space travel. 

This article will give you an idea of calculus, explain the evaluation of definite integrals and give information about its uses and applications.

What are 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 the evaluation of definite integrals, we can take limit points as (a,b) and find the area of the curve with respect to the x-axis.  

We 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 the addition of areas, definite integrals are the sum of areas with definite limits. 

Properties of Definite Integrals

Definite integrals have the following properties:

• The integral contains no area and is a line if the limits are the same in the integration. Limits, when reversed, should be started with a negative sign in front of the integral. 
• The sum of the integrals and integral of an aggregate are the same. It means that the sum of integrals of every function is the integral of the sum of two functions. Plus and minus signs indicate this rule works to calculate the difference between the two functions.
• The integral of the difference of two functions and the difference of the integrals are equal. This property is called the difference rule of integration, and we can show it as an equation in mathematical form.

How Can You Calculate Definite Integrals?

The first step towards the evaluation of definite integrals is to calculate the antiderivative F(x)

The second step towards it is to calculate the values of F(b) and F(a)

The third step is to calculate F(b)-F(a)

Example.

• Question 1)       ∫²₁ 2x dx
  • At x=1: ∫2x dx = 12 + C
  • At x=2: ∫2x dx = 22 + C
  • (22 + C) − (12 + C)
  • 22 + C − 12 − C
  • 4 − 1 + C − C = 3
• ∫²₁ 2x dx = 3

A = 2+4 × 1/2 = 3

It has an area of 3.

Definite Integral Application

There are various applications of definite integrals, some of which are as below:

• We can calculate the area of a circle through the evaluation of definite integrals. First, we determine the area of the circle portion in the first quadrant. Then, we multiply this value by 4 to get the total area of the circle.
• We can find the area of a parabola using the evaluation of definite integrals. A parabola’s axis divides it into two symmetric sections. We initially find the area of the parabola in the first quadrant under the x-axis using the evaluation of definite integrals. Then, we multiply the area by 2 to get the total area of the parabola.
• We can use the evaluation of definite integrals to calculate the bounded area of an ellipse in the first quadrant concerning the x-axis. This area is then multiplied by 4 to get the total area of the ellipse.

Conclusion

Calculus is a branch of mathematics related to the instantaneous rates of change in matter and the aggregation of small factors that are infinite. We use calculus to understand concepts of time, space and motion. We can develop calculus by working with small quantities. 

This article gave you an idea of the evaluation of definite integrals and their uses and applications.