Integration is the most important technique used to determine various factors like area under a curve and point equations etc. basically, integration is the process of determining a function g(x) whose derivative is the real function f(x). Many techniques have been introduced in integral calculus to resolve complex problems. Along with that, an online Riemann sum calculator by calculator-online.net also helps you to determine the area under the curve instantly. 

 

Well, in this article, I will be discussing the method to find the area under a graph using definite integrals. 

Stay focused:

The Area Under A Graph:

You can easily determine the area under a graph with the help of a definite integral. A definite integral executes a function of a curve for resulting the area bounded by it. Also, a Riemann sums calculator also provides here another edge to resolve complex curves is not it a good thing for you? Yes, it is! If you are a student, then you must make free use of this right Riemann sum calculator. Let m,e guide you on how to determine a definite integral.

Definite Integral:

An integral with upper and lower limits that are defined is known as the definite integral. This integral is always defined on an interval [a, b].

The General Form Of Definite Integral:

Suppose you have a function that is defined for the said interval. Now you divide the interval into its corresponding subinterval of length \(\deltax\) and make a selection of the point \(x_{i}^*\). In this way, the definite integral become:

 

$$ \int_{{\,a}}^{{\,b}}{{f\left( x \right)\,dx}} $$

$$ = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {f\left( {x_i^*} \right)\Delta x} $$

 

Lower Limit:

The subscript a which is at the bottom of the integral represents the lower limit of the integral.

 

Upper Limit:

The constant b represents the upper limit of the integration.

 

Interval Of Integration:

The area among a and b are collectively termed as the interval of the integration.

 

Here the free Riemann sum calculator also helps you to determine the limits of the function to solve for area under the triangle in the form of finite sums.

Example:

Determine the area under the curve that is represented by the following equation:

 

$$ y = \int_{{\,1}}^{{\,3}}{{x^{2}}} $$

 

Solution:

We know that:

 

$$ y = \int_{{\,a}}^{{\,b}}{{x^{2}}} $$

$$ y = |\frac{x^{2+1}}{2}|_{1}^3$$

$$ y = \frac{1}{2}|x^{3}|_{1}^3 $$

$$ y = \frac{1}{2}\left(3^{3} – 1^{3}\right) $$

$$ y = \frac{1}{2}\left(9-1\right) $$

$$ y = \frac{1}{2}*8 $$

$$ \frac{8}{2} $$

$$ y = 4 $$

 

Which represents the area under the curve that is defined by the function given. In case it gets difficult for you to determine the area, try using the free triangle sum theorem calculator.

Wrapping It Up:

In this article, I discussed how to evaluate a definite integral for finding the area under a graph. Also, the use of a free Riemann sum calculator has also been highlighted so that you may get the advantage of instant calculations at one single tap. I hope it will help you people a lot!