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Integral Definition

As per integral definition, It is the assigning of numbers to define & describe area, volume, displacement & other concepts. In calculus, integration is the most important operation along with differentiation.

What is Integration?

Integration is used to find differential equation of an integral or integrals. It produces a function to differentiate and calculate the area under the curve of a graph of the function.

Integral definition is used to find areas, central points, volumes etc. Integration calculator is used to define integral to find the area under the curve like this:

Integral Calculator


F(x) is the function and

A is area under the curve.


Click on to learn formula & the calculations of Variance & Covariance.


What is Integrand?

It is a function which is to be integrated is called Integrand. In an integral equation or integration formula, it is denoted as the function f(x).

What is Integral Notation?

For an integral equation


∫ is the Integral Symbol and 2x is the function we want to integrate.

In this integral equation, dx is the differential of Variable x which highlights that the Integration's variable is x. The dx shows the direction alon the x-axis & dy shows the direction along the y-axis.

Integral symbol & integral rules are used by integrals calculator to get results quickly. Learn more about the scientific notation from here.

How to calculate Integral?

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)

calculate integral

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)

How to calculate integral?

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

inverse of derivative

What is Definite Integral?

The definite integral has both the start value & end value. The definite integral of function f(x) represents the area under the curve from x = a to x = b.

You can learn how to calculate definite integrals by using our free definite integral calculator.

What is Indefinite Integral?

The indefinite integral does not have the upper limit and the lower limit of the function f(x). The indefinite integral is also known as antiderivative

Use our indefinite integral calculator to solve definite & indefinite values. Learn how to find limit of function from here.

What is Double Integral?

Double integral have 2 variables. Consider f(x,y) as a function in a 3D space in xy-plane and R be any region in xy-plane. If 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

Use Trapezoid area calculator & Rectangle area calculator to further strengthen your math concepts related to area & surface.

Double Integral Formulas

Below are the 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

Try Quadratic formula calculator & Distance formula calculator to learn about different math formulas used to solve different math related equations.

How to calculate Double Integrals?

One difficulty in the computation of double integrals is to determine the limits of integration. The limits of integration as dxdydxdy order are required to determine the limits of integration for the equivalent integral dydxdydx order.

In double integrals, the process of switching between dxdydxdy order and dydxdydx order is called changing the order of integration. The integral calculator with limits helps you to get accurate results.

How to use Integral Calculator with steps?

Our integral calculator is the best integration by parts calculator.

how to calculate integral

Just enter your equation like 2x+1

Press "CALCULATE" button and the Integral Calculator will calculate the Integral of the function step by step. You can further check these steps also.

Integral Calculator

You can also use our other free calculators like Standard Deviation Calculator & Cross Product Calculator for free.

Please provide your valuable feedback below. Best of luck with your learning and calculations. Cheers!

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