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Evaluate the Definite Integral Calculator + Online Solver With Free Steps
A Definite Integral Calculator is used to calculate the definite integral of an algebraic expression, where Algebraic Expressions are used to represent real-world problems in the form of a mathematical model.
This calculator comes in very handy for solving definite integrals as it takes away the rigorous procedure involved in solving them by hand.
What Is a Definite Integral Calculator?
A Definite Integral Calculator is an online calculator that solves mathematical models’ definite integrals.
Definite Integrals represent a type of integration where the upper and lower bounds for integration are known. Therefore, they provide a definite solution to whatever problem you apply them.
They are often applied to trigonometric equations, algebraic equations, and so on, and they are very commonly used in the field of Engineering and Physics. They can be applied to mathematical models to find shapes of buildings and centers of gravities of objects.
How to Use a Definite Integral Calculator?
A Definite Integral Calculator can be used by entering your mathematical queries into the input boxes provided and then pressing the “Submit” button. The step-by-step process for getting the best results from this calculator is given below.
Step 1
You can start by setting up the problem you would like to find the definite integral for and entering the expression into the text box labeled “Integrate.”
Step 2
Following the setup and entry of the expression, you enter the variable and the upper and lower bounds of the integral are labeled as “From,” “=,” and “to,” respectively.
Step 3
Once you have entered all the required values into the text boxes, you can now press the “Submit” button. This will solve your problem and provide you with a solution in a new window.
Step 4
Finally, if you intend to solve more problems of that sort, you can enter those problem statements in the input boxes. This can be done in the new pop-up window.
An important fact to notice is that this calculator is designed to work for only one variable’s integration at a time.
How Does a Definite Integral Calculator Work?
A Definite Integral Calculator works by solving the definite integral for the input mathematical expression relating to any function. These functions can be of any form involving a certain variable, trigonometric, algebraic, etc.
What Is Integration?
Integration is the mathematical process of piecing together infinitesimal data to define concepts such as volume, displacement, etc. In mathematics, Integrals correspond with the act of allocating values to functions.
Integration is widely used in Engineering, Mathematics, and Physics. They help in acquiring results of areas under curves of different types of functions and to find significant features of three-dimensional objects.
What Is a Definite Integral?
A Definite Integral is a type of integral in which the limits of the integration are known. The Limits of Integration describe the resulting function’s definition region in space and time.
The basis of Physics and Physical Laws and theories are based on this calculus. Definite Integrals are used to calculate work functions, power, mass, etc. because a definite integral provides a definite result as a particular integral is valid in a specific region or boundaries.
How To Calculate a Definite Integral
To calculate a Definite Integral, you will first require a function on which you intend to calculate the integral. Then, you will need the variable you would integrate the expression with so you can apply limits to this integration problem.
The difference between a regular and definite integral doesn’t show until the integration is done. This Integration takes place according to the rules of integration, set in place for all sorts of variables and their combinations.
Once the integral has been solved for a variable, then a limit is applied to the resulting expression. This limit, when defined such as in a Definite Integral problem, can provide a definite result to the given problem.
Solving the Limit
Solving the limit involves a sum of values of the integration result. So if you have a problem of this type:
\[ \int_{a}^{b} f(x) \,dx = g(x)\]
And after you have a resulting $g(x)$ function, it has to be solved as such:
\[ \int_{a}^{b} f(x) \,dx = g(x) \bigg \vert \begin{matrix}b \\ a\end{matrix} = (g(b) – g(a)) = y\]
Where y represents the resulting definite solution corresponding to the original problem f(x).
History of Definite Integrals
Definite Integrals, like so many other powerful mathematical operations, have an interesting history associated with them. They are believed to have been used back in even the ancient Greek era.
But the modern-day integration stems from the work brought forward by Gottfried Wilhelm Leibniz and Isaac Newton during the 17th century, where the area of a curve was broken down and expressed mathematically as a sum of an infinite number of rectangles having an infinitesimally small size.
Another big name in the field of Integration and Calculus is indeed Bernhard Reimann, known for his famous Reimann’s sum.
All of these integrations trace back originally to the oldest known method of finding areas, the Method of Exhaustion. This method relied on breaking any unknown area of a shape down into several objects for which the area was known. This method dates back to the days of Ancient Greece.
Solved Examples
Here are some examples regarding this concept and this calculator.
Example 1
Consider the given function f(x) = sin(x)
Solve a definite integral for this function corresponding to x ranging from 0 to 1.
Solution
Now applying a definite integral on this function gives us:
\[ \int_{0}^{1} \sin (x) \,dx = – \cos (x) \bigg \vert \begin{matrix} 1 \\ 0 \end{matrix} = 1-\cos (1) \approx 0.45970 \]
Example 2
Consider the given function f(x) = 2x
Solve a definite integral for this function corresponding to x ranging from 1 to 2.
Solution
Now applying a definite integral on this function gives us:
\[ \int_{2}^{1} 2x \,dx = x^2 \bigg \vert \begin{matrix} 2 \\ 1 \end{matrix} = 3 \]