Definite integrals are the integrals that are expressed as the difference between the values of the upper and lower limits of an independent variable. Integration is useful in finding the area, volume, and several other things. It is a method of finding the area beneath the function and axis using stripes of the region. A definite integral for a real-valued function ‘f’ and real Numbers (a<b) can be written as:
The equation represents the area between the function f, x-axis, and the lines x=a & x=b. The region above the x-axis represents the positive area, while the region below the x-axis will represent the negative area.
You can understand the concept better with the help of this picture above. The blue region is the ‘area’ which this definite integral represents.
Definite Integration: Easily Explained with Examples
Now let us take an example so that we can understand this concept well.
Let us begin by performing integration of x^2 in the usual way. To showcase the problem of a definite integral, the result is in square brackets. The limits of integration are written on the upper and lower corner of the right bracket as below:
Now, the evaluation of the value in the square brackets starts, by applying rules, first by placing the value of the upper limit in x, then by placing the value of the lower limit in x. Subtraction of these two individual results gives the value of the definite integral.
After the step mentioned above, we have to
(Evaluate the upper limit) – (Evaluate the lower limit)
Note, the constants of integration cancel each other out. On its occurrence, we can ignore them when we are evaluating definite integrals in the future.
Points to Remember while Solving Definite Integrals
- The Definite integral of 1 is always equal to the interval between the limits.
- A constant factor can be moved outside of the integration sign.
- If the upper limit and the lower limit of the integral are the same, then the result is zero.
- A Definite integral of the sum of two functions is equal to the sum of definite integral of both the functions when solved individually.
- A Definite integral of the difference of two functions is equal to the difference of definite integrals of both functions when solved individually.
- Suppose there is a point c in between interval a and b, then the Definite integral of function f over the interval (a,b) is equal to the sum of integral over the intervals (a,c) & (c,b).
- Definite integral of any non-negative function is always greater than or equal to zero.
- Definite integral of any non-positive function is always less than or equal to zero.
- We can interchange the position of both the limits, but after the interchange, we must add a negative sign to the integration problem.
- We can use the definite integral to find the area between two curves.
Some Additional Things to Remember
For definite integrals, the values may occur as a result that is positive or negative or maybe even zero. Although, we should always keep it in mind that ‘Area‘ can never be Zero, as ‘Area‘ is a non-negative segment. The upper limit and the lower limit can also have similar or opposite signs. The only difference between definite and indefinite integrals are its limits.
The limits bound a specific area, as shown in the picture displayed above in this article. Here, the investigation of continuity of interval is by definite integral and the discontinuity of interval is by indefinite.
There are various properties which can make the complexity of problems more manageable. You should mark and memorize these fundamental rules like the ‘Simpson’s Rule‘ and the ‘Trapezoidal Rule’ for future reference as these two rules prove the most efficient in solving a definite integral problem.
Now let us take another example,
This determines the relationship between integrals and derivatives.
The above mentioned examples are the types of questions under the definite integral section of calculus, and to master it, you have to start practicing. Start with the very basics, which is the type of questions you see above, then start stepping up the level as progressive difficulty always helps you to understand a concept better.
So go ahead and try this and see for yourself that definite integral is a fascinating topic to solve. We hope that you understood the examples and also all the important properties. These properties will help you to categorize a question properly before approaching it.
All the Best!
Also Refer: What is Continuous Integration?