Properties of Definite Integrals

Introduction:

Integration is the process of bringing two or more items together. In mathematics, integration refers to combining functions. Alternatively, integration may also be referred to as summation since it is used to summarise the whole function or, in a visual sense, it is used to compute the area under the curve function when a function is shown as a curve.

Evaluation of integrals has traditionally been conceived of as the opposite of operation differentiation, breaking a function into smaller functions. At the same time, integration brings the smaller pieces together to form a total area under the curve. In this article, we will explain definite integral properties in detail.

What is integration?

First, one must establish how much space is accessible under the curve that must be addressed before calculating the function’s integrals. This is accomplished by drawing as many little rectangles as feasible to cover the whole area and then merging the regions of those rectangles. It is necessary to integrate a function to derive its antiderivative. A function must have finite boundaries on the integration domain to qualify as definite integration.

Definite integral

The curve area can be calculated using the definite integral method. A curve’s starting and ending points compute the area under a curve. For example, [a, b] can be used to calculate the area of the f(x) curve if the x-axis is considered. The related definite integral expression is  ∫ba f(x)dx. You can use definite integrals to know how big an area is. For example, to find the area of a circle, parabola, or ellipse.

Definite integral formula

A definite integral is evaluated using definite integral formulae. We can use two formulae to determine the value of a definite integral. The ‘basic theorem of calculus’ and the ‘definite integral as a limit sum’ are the names given to the first and second formulae, respectively. 

• ∫baf(x)dx = limn Σr=1n   hf(a+rh),

where h= b-a/n

• ∫baf(x)dx = F(b)-F(a), where F'(x)= f(x)

Properties of definite integrals

For solving integral problems, the definite integral properties provide a framework. Based on the types of integrals, the properties of integrals can be broadly divided into two groups: indefinite integrals and definite integrals. In this article, we will discuss the properties of definite integrals.

These properties are helpful to integrate the given function and apply the lower and the upper limit to find the value of the integral. Let us check below some important properties of definite integrals.

Properties of definite integrals are :

• ∫baf(x)dx = ∫baf(t)dt

• ∫baf(x)dx = -∫abf(x)dx

• ∫ba cf(x)dx = c∫baf(x)dx

• ∫ba (f(x)±g(x))dx = ∫baf(x)dx±∫bag(x)dx

• ∫baf(x)dx= ∫caf(x)dx+∫bc f(x)dx

• ∫baf(x)dx=∫baf(a+b-x)dx    [Kings rule]

•  ∫a0f(x)dx=∫a0f(a-x)dx 

• ∫2a0f(x)dx=2∫a0f(x)dx  if f(2a-x)= f(x) 

• ∫2a0f(x)dx=0  if f(2a-x)= -f(x) 

• ∫a-af(x)dx=2∫a0f(a)dx if f(x) is an even function

• ∫a-af(x)dx=0 if f(x) is an odd function and f(-x) = – f(x)

Limit sum of definite integral

As we saw in the last section, the area under a curve may be written as the sum of an infinite number of rectangles. The precise integral may be determined using ∫baf(x)dx.

Each subinterval of [a, b] is subdivided into a different number of rectangles to get an infinite number of rectangles. In this way, the definite integral’s limit sum formula is as follows:

∫baf(x)dx =  limn Σr=1=hf(a+rh),

Here h = (b-a)/n is the length of each subinterval which is infinitesimally small when n tends to infinity.

This is an indication that a infinite sum may be expressed as an integral.

Conclusion

Properties of definite integrals help us solve complex definite integrals with ease. They simplify the problem rather than making us go through rigorously solving the integrals which is not possible in some cases.