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Double integral Calculator

Double Integral Calculator

Enter a function to Integrate f(x, y): ?

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An online double integral calculator with steps 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!

What is Double Integral?

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.

Double Integration Rule:

Here we discuss some important formulas and rules that are used by double integral calculator 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:

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

Properties of Double Integration:

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 

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

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

Double Integral Area:

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 Integral Calculator allows you to evaluate the integrals of the given functions with respect to the variable involved.

Double Integration in Polar Coordinates:

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.

How to do Double Integrals?

Here’s the complete procedure for solving double integrals that are used by the 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 compute the double integration 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:

Find the double integration for x^2 + 3xy^2 + xy with limit values (0, 1) for x and y variable.

Solution:

The two variable 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, double integral calculator 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, 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:

$$ ∫ y dy = y^2 / 2 $$

Then,

$$ =y^3 + y^2 / 2 $$
$$ = y^3 / 2 + y^2 / 4 $$
$$ = xy / 3 + y^3 / 2 + y^2 / 4 $$
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 adds the constant of integration:

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

The answer is:

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

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

How Double Integral Calculator with steps Works?

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

Input:

  • 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.

Output:

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

FAQs:

What are the applications of double integral?

  • 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.

Does order have any effect on double integration?

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

Is it possible to split the double integral?

Fubini’s Theorem states, “we can split up the double integrals into some iterated integrals”.

Conclusion:

Use this online double integral calculator 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.

Reference:

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

From the source of Lumen Learning: Antiderivatives, Area and Distances, Finding arc length by Integrating, The Fundamental Theorem of Calculus, Indefinite Integrals and the Net Change Theorem.

 

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.