We now understand the relationship between differentiation and integration.These are inverse operations in the sense that integration requires taking the antiderivative of a function.
For basic functions this is easy to do, but it gets extremely complicated as the functions get more complex, and we will have to learn a number of different strategies to tackle the tougher ones but that will come later.
For now let's just go over a few properties of definite integrals which will be helpful in examples. But Before it we will take an overview that what is definite integral.
Definite Integral
The definite integral tells us the exact area under the curve. In formal way, definite integrals are defined as:
Consider we have a function f(x) which is defined in an interval [a,b], then the definite integral of that function from point a to b is given as:
$$ \int_a^b f(x) \; dx = \lim\limits_{n \to ∞} \sum_{i=1}^n \; f(x_i^*) Δx $$
Definite Integral Notation
As we can see, the notation for indefinite integral is almost as it is used in definite integral notation. But there is a little bit of change that we can analyze that there are two numbers “a” and “b” at integrals.
The number “a” that is available at the bottom of the integral sign is known as the lower limit of the integral while the number “b” that is available at the top of the integral sign is known as the upper limit of that integral .
Basically these upper and lower limits are called the interval of that definite integral because this integral is defined on the following interval of [a,b]. The general form of definite integral notation is given below.
$$ \int_a^b f(x) \; dx = F(b)-F(a) $$
Here you can see the basic integral rule for definite integral. Every definite integral requires to go through that rule while evaluating definite integral because it is the core part of definite integral properties.
Properties of Definite Integral
Definite integral properties are the basic rules of integral which are generally used when we are evaluating definite integrals in integral problems. Definite integral is a complex process while evaluating but it become simpler if we follow some certain definite integral properties with steps to solve it.
As we said it will be a good idea to quickly highlight some important definite integral properties. So, let's see what are these rules of integral which makes integration easy and understandable. These properties of integral are given as:
- First, if we have some integral f(x) over the interval from [a,b], then if we switch the limits of integration, integrating instead from [b,a], this will be the same as the first integral but negative. Because we are integrating in the opposite direction.
$$ \int_a^b f(x) \; dx = - \int_b^a f(x) \; dx $$
- If both of the limits of integration are the same number the integral will be equal to zero.
$$ \int_a^a f(x) \; dx = 0 $$
No matter what the function is, this is because we are essentially asking about the area under a single point which is just a line. A line has no area, so this integral equals zero.
- Similarly if we have two integrals of the same function f(x) over adjacent intervals one from “a to b” and the other from “b to c”. The sum of these integrals is equal to the integral of the function over the whole interval from “a to c”.
$$ \int_a^b f(x) \; dx \;+\; \int_b^c f(x) \; dx = \int_a^c f(x) \; dx $$
This is the same as saying that the area of the first section plus the area of the second section is equal to the total area under the curve.
- When integrating a constant we will simply get that constant times the quantity of the upper limit minus the lower limit. Mathematically,
$$ \int_a^b C \; dx = C(b-a) $$
- Similarly if we are taking the integral of a constant times some function f(x), we can just pull the constant out of the integral and we get the constant times the integral of the function.
$$ \int_a^b c \times f(x) \; dx = c \int_a^b f(x) \; dx $$
- We also have the property that says that the integral of a sum of functions f(x) over some interval [a,b] is equal to the sum of their integrals over the same interval [a,b].
$$ \int_a^b \left[ f(x) \;+\; g(x) \right] \; dx = \int_a^b f(x) \; dx \;+\; \int_a^b g(x) \; dx $$
the same goes for the difference of functions we just change these plus signs to minus signs.
$$ \int_a^b \left[ f(x) \;-\; g(x) \right] \; dx = \int_a^b f(x) \; dx \;-\; \int_a^b g(x) \; dx $$
- Let's also quickly note that if the interval of a function that is being integrated is above the x-axis its integral will be positive.
$$ \int_a^b f(x) \; dx \gt 0 $$
- If instead it is below the x-axis, it will be negative, as these are negative y-values.
$$ \int_a^b g(x) \; dx \lt 0 $$
- If the function crosses the x-axis with some portion above and some portion below, the integral will be the net area or the area above the axis minus the area below it.
$$ \int_a^b f(x) \; dx = \text{Area above axis} - \text{Area below axis} $$
So these are some useful and helpful properties of definite integrals which are required for evaluating the definite integral in our practical example problems.