# Integral Calculator

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## Integral Definition

Integral is a given function in the derivative, i.e. which produce that function when we differentiate it, and calculate the area under the curve of a graph of the function.

Integration is used to find areas, central points, volumes, and many useful and important things, Integral Calculator will help you to find integral. But it is very simple to start with the function to find the area under the curve like this:

Where,

F(x) is the function and

A is area under the curve.

## How to calculate integral?

We can calculate the function at a few and simple steps first divide the area in slices and add up the width of these slices of Δx. Then the answer won't be accurate. (look at figure 1)

If we make Δx a lot in smaller width and add up all these small slices then the accuracy of the answer is getting better. (look at figure 2)

If the width of the slices approach zero, then answer approaches to the true or actual result. So,

We now say that dx to mean the Δx slices are approaching zero in width.

Note that the integral is the inverse of derivative

## Notation

Integral is represented by a stylish "S" and “s” is the sum of all the slices to find the area under the curve.

$$∫2xdx$$

Where, ∫ is the Integral Symbol, 2x is the function we want to integrate and at last we have dx that show us the direction along X-axis. (if there is dy then this show us the direction along Y-axis).

First, we use Integral Symbol then we put the function in Integral Calculator we want to calculate the integral and it is also called the Integrand.

then at the end we put dx to indicate that the slices go in the x-axis direction and in width it’s approach to zero.

And here is how we write the answer:

$$∫2x+1dx$$

## Definite Integral

The integral is outlined to be precisely the limit and summation that we tend to check to find the net area between a function and the x-axis. the structure for the definite integral is very similar to the notation for an indefinite integral.

There is a little bit of terminology we need to understand. The value “a” that is at the bottom of the integral sign is called the lower limit of the integral and the value “b” at the top is called the upper limit of the integral.

## Definition of Double Integral

Following phenomena can help us to understand the definition of Double integral more easily:

Consider f(x,y) as a function in a 3D space in xy-plane and R be any region in xy-plane.Let we divide R region into smaller sub-regions and δAi=δxi δyi be the area of its sub-region.Then,the double integral of f(x,y) over the region R can be defined as:

ʃʃR ƒ (x, y) dxdy=limn ͢ 0 Σ (n, i=1) ƒ (xi, yi) δxi δyi

where (xi, yi) is any point in the ith sub-region

## Double Integral Formulas

Here is the list of some Double Integral Formulas with different functionalities:

If f (x, y) ≥0 in a region R and S⊂R:
∬Sf (x, y) dA≤ ∬Rf (x, y) dA

• To find double integral of sum of two functions:
∬R [f (x, y) +g (x, y)] dA= ∬Rf (x, y) dA +∬Rg (x, y) dA

• To find double integral of difference of two functions:
∬R [f (x, y) −g (x, y)] dA= ∬Rf (x, y) dA −∬Rg (x, y) dA

• In case of a constant factor:
∬Rkf (x, y) dA= k∬Rf (x, y) dA

• For f (i, n) ≤ g (x, y) in R:
Rf (x, y) dA≤ ∬Rg (x, y) dA

• To find the volume of a solid:
V=∬Rf (x, y) dA

• To find volume of solid between two surfaces:
If f (x, y) ≥ g (x, y) over a region R, then:
V=∬R [f (x, y) −g (x, y)] dA

Sometimes, Double integrals become easy to be evaluated in case, we change the order of integration or when we change the polar coordinates over regions to which polar equations provide boundaries. Generally, change of variables includes evaluation of multiple integrals by substitution. The main purpose of substitution is to replace complicated integrals by one that are easier to evaluate.

One difficulty in the computation of double integrals is to determine the limits of integration. In some cases, we know the limits of integration as dxdydxdy order and required to determine the limits of integration for the equivalent integral in dydxdydx order or vice versa. In double integrals, the process of switching between dxdydxdy order and dydxdydx order is usually called changing the order of integration.

Changing the order of integration is a little bit tricky as it is difficult to write a specific algorithm for this procedure. This task can be made easy to be accomplished by drawing a picture of the radion D. By picture, you will be able to determine the corners and edges of region Which is what you need to write down the limits of integration.

## Integral Calculator

Calculatored introduced Integral Calculator to calculate the Integral functions: (Calculatored, 2019)

Where,

You have to enter equation,

After you entered the equation then press “CALCULATE” button and the Integral of the function is calculated.

Here is an example to calculate the Integral of the function through calculatored Integral calculator.

After entered the equation 2x+1

Press “CALCULATE” button and the Integral Calculator will calculate the Integral of the function on the right side of the Integral Calculator in the status block.