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**Table of Content**

An online double integral calculator with steps free helps you to solve the problems of two-dimensional integration with two-variable functions. The calculation of two consecutive integrals enables you to compute the function areas with two variables to integrate over the given intervals. Here you can learn how to solve double integrals and much more!

Double integral of function f (x, y) over the rectangular plane S in the xy plan is expressed by \( ∫∫_S f(x, y) dA = lim _{j,k -> ∞} ∑^m_{I = 1} ∑^n_{j = 1} f(y_{ij}, x_{ij})△A \).

It is mainly used to determine the surface region of the two-dimensional figure, which is donated by “∫∫”. By double integration, we can find the area of the rectangular region. If you have good knowledge about simple integration, then it will be very easy for you to solve the problems of double integration. So, begin with some basic rules of double integration.

Here we discuss some important formulas and rules that are used by double integral calculator over region to perform double integration. For solving the integration problems, you have to study different methods such as integration by substitutions and integration by parts or formulas. In the double integrals, the rule for double integration by parts is mentioned below and also taken into consideration by this best double integration solver while carrying out calculations.

$$ ∫∫m dn/dx dx . dy = ∫[mn -∫n dm/dx dx]dy $$

Here are some important properties of double integral:

$$ ∫_ {x = a} ^b ∫_ {y = c} ^d f (x, y) dy . dx = ∫_ {y = c} ^d ∫_ {x = a} ^b f (x, y) dx . dy $$

$$ ∫∫( f(x, y) ± g (x, y)) dA = ∫∫ g (x,y) ± dA ∫∫ f(x, y) dA $$

**If f(x, y) < g(x, y), then ∫∫g (x, y) dA > ∫∫ f(x, y) dA **

$$ k ∫∫ f (x, y) . dA = ∫∫ k. f(x ,y). dA $$

$$ ∫∫ R ∪ S f (x, y) . dA = ∫∫ R f (x, y). dA + ∫∫ sf (x, y). dA $$

When we need to find the double integration of variable M, let M = f(x, y) define over the domain K in the plan of xy. If we find the endpoints for x and y as the limits of region and divide the certain region into the vertical stripes, then we use the formula:

$$ ∫∫_K f(x, y) dA = ∫^{x = b} _{x = a} ∫^{y = f_2 (x)} _{y = f_1 (x)} f(x, y) dy dx $$

If the function m is the continues function, then:

$$ ∫∫_K f(x, y) dA = ∫^{x = b} _{x = a} ∫^{y = f_2 (x)} _{y = f_1 (x)} f(x, y) dy dx = ∫^{x = d} _{x = c} ∫^{x = n_2 (y)} _{x = n_1 (y)} f(x, y) dx dy $$

However, an online multi Integral Calculator allows you to evaluate the integrals of the given functions with respect to the variable involved.

In polar coordinates, the double integration is:

$$ ∫^{θ_2} _{θ_1} ∫^{r_2} _{r_1} f (r, θ) dθ, dr $$

First, we must have to integrate the f(θ, r) with respect to r between the limits \( r_1 and r_2 \), where θ is constant and integrate the resulting equation as θ from \( θ_1 to θ_2, \text { where } r_1 and r_2 \) are constant. And to resolve such queries, you can also use the online polar coordinates double integral calculator.

Here’s the complete procedure for solving double integrals that are used by the polar double integral calculator with steps. However, you can do double integration manually by following steps:

- First, take the function with x and y variables.
- Now, perform the integration with respect to y to solve double integration online of the function.
- Then, add the range values of (a, b) in the certain function for getting f(a) and f(b).
- Finally, perform the integration one more time for other variables and substitute the range values again for obtaining the f(a) and f(b).

**Example:**

**Evaluate double integral x^2 + 3xy^2 + xy with limit values (0, 1) for x and y variable.**

**Solution:**

The two variable multiple integral calculator provides the Indefinite Integral:

$$ x^2y (4x + 6y^2 + 3y) / 12 + constant $$

Also, the double definite integral calculator displays the definite integral for the given function as:

=13 / 12

**Integral Steps:**

First, we take inner integral:

$$ ∫ (x^2 + 3xy^2 + xy) dx $$

Now, the double integral solver Integrate term-by-term:

The integral of \( x^n is x^{n+1} / n+1 \) when n≠−1:

$$ ∫x^2 dx = x^3 / 3 $$

$$ ∫ 3xy^2 dx = 3y^2 ∫ x dx $$

The integral of x^n is x^{n+1} / n+1 when n≠−1:

$$ ∫x dx = x^2 / 2 $$

So, the result is:

$$ 3x^2y^2 / 2 $$

$$ ∫xy dx = y ∫ x dx $$

The integral of x^n is x^{n + 1} / n+1 when n≠−1:

$$ ∫x dx = x^2 / 2 $$

So, the result is: \( x^2y / 2 \)

Then,

$$ x^3 / 3 + 3x^2 y^2 / 2 + x^2y / 2 $$

Now, the free definite double integral calculator polar simplifies:

$$ X^2 (2x + 9y^2 + 3y) / 6 $$

The double integrals calculator substitutes the constant of integration:

$$ X^2 (2x + 9y^2 + 3y) 6 + constant $$

So, the answer is:

$$ X^2 (2x + 9y^2 + 3y) 6 + constant $$

Then we take second integral:

$$ ∫x^2 (x^3 + y(3y + 1) / 2) dy $$

$$ ∫x^2(x^3 + y(3y + 1) / 2) dy = x^2∫ (x^3 + y(3y + 1) / 2) dy $$

The second integral calculator again perform integration term-by-term:

The integral of a constant is the constant times the variable of integration:

$$ ∫ x^3 dy = xy^3 $$

$$ ∫y(3y + 1)^2 dy = ∫y(3y + 1) dy^2 $$

Now, the step by step double integral calculator rewrites the integrand:

$$ y(3y + 1) = 3y^2 + y $$

Now, the double integral volume calculator Integrates term-by-term:

$$ ∫3y^2 dy = 3 ∫y^2 dy $$

The integral of \( y^n is y^{n+1} / n+1 \) when n≠−1:

$$ ∫y^2 dy = y^3 / 3 $$

So, the result is: \( y^3 \)

The integral of \( y^n is y^{n + 1} / n + 1 \) when n≠−1:

$$ ∫ y dy = y^2 / 2 $$

We know you are bored of these complex calculations, but do not worry! This best double integral over region calculator can do this all for you in seconds and accurately. Anyways, moving ahead further:

$$ =y^3 + y^2 / 2 $$

$$ = y^3 / 2 + y^2 / 4 $$

$$ = xy / 3 + y^3 / 2 + y^2 / 4 $$

So, the result is:

$$ X^2(xy / 3 + y^3 / 2 + y^2 / 4) $$

**Now simplify:**

$$ X^2y (4x + 6y^2 + 3y) / 12 $$

Then, the double integration calculator with steps adds the constant of integration:

$$ X^2y (4x + 6y^2 + 3y) / 12 + constant $$

The answer is:

$$ X^2y (4x + 6y^2 + 3y) / 12 + constant $$

Apart from the manual computations that seem to be tricky enough, you must cross check and speed up your calculations with this iterated integral calculator with steps.

However, an online Triple Integral Calculator helps you to find the triple integrated values of the given function.

An online double Integral solver determines the double integral of a given function with x and y limits by following these steps:

- First, enter a function to integrate and select the intervals from the drop-down list.
- If you know the limits, then select definite and substitute the definite values for upper and lower limits.
- On the other hand, if you don’t know the limit values of a certain function, select the indefinite.
- Hit the “Calculate Double Integral” button.

- The double integrals calculator displays the definite and indefinite double integral with steps against the given function with comprehensive calculations.

- It is used to determine the area of the region bounded by the curve of a particular function.
- It is mostly used to find the undersurface volume.
- Double integrals are used to compute the average value of the two-variable function.

Generally, the order of double integral does not matter. If important, then you should rewrite the iterated integral, when you change the integration order.

Fubini’s Theorem states, “we can split up the double integrals into some iterated integrals”. For further assistance, you may let this fubini’s theorem calculator split the integrals and display the whole scenario of calculations involved in the process.

The conversion of polar coordinates in another system is done by following three ways listed below:

- You need to map the rectangle D in the left plane
- As well, you also need to map the rectangle D to the region D in the (x, y) plane.
- To speed up your conversion speed, you can let this double polar integral calculator do that for you.

Use this online evaluate double integral calculator step by step that provides the resultant values for both definite and indefinite double integrals in any order using the algebra system. In symbolic integration, the double integral solver utilizes the integral and algebraic rules for taking the antiderivative of the given function before applying the calculus fundamental theorem for double integration.

From the source of Wikipedia: Multiple integral, Riemann integrable, Methods of integration, Integrating constant functions, Use of symmetry.

From the source of Libre Text: General Regions of Integration, Double Integrals over Non-rectangular Regions, Fubini’s Theorem (Strong Form), Changing the Order of Integration, Calculating Volumes, Areas, and Average Values, Improper Double Integrals.