Double Integral Calculator

Double Integral Calculator:

This double integral calculator helps you evaluate definite or indefinite double integrals of two-variable functions (f(x, y)). The double integral solver provides step-by-step calculations and even allows you to change the order of integration, leading to an easier solution.

What Is Double Integral?

In Calculus, a double integral is used to compute the integrals of two variable functions ( denoted by f(x, y)) over a two-dimensional region (denoted by R). It not only helps to find the volume under surfaces but also the mass distribution, and compute flux (rate of flow) and area over the region\(\ R^{2}\ \).

A double integral is mathematically represented by the symbol \(\ ”∫∫_R”\), which indicates a double integral over region "R" followed by the function f(x, y) and the area element dA.

A double integral can also be shown as an iterated integral:

\(\begin{array}{l}\ ∫∫_{R}f(x,y)\ dA =\ ∫∫_{R}f(x,y)\ dx\ dy\end{array}\)

How To Solve Double Integrals?

To calculate the double integral of the 2-dimensional functions, follow these steps:

• First of all, specify the region (denoted by R)
• Now, write the double integral in the notation form:\(\ ∫∫_R f(x, y) dA\)
• Perform the inner integral on the function f(x, y) for one variable and treat the second variable as a constant
• Write down the result and integrate it into the second variable, keeping the first variable as a constant
• You will have the final result representing the double integral evaluated over the region R

Example:

Evaluate double integral \(\ x^{2}\ + \ 3xy^{2}\ + \ xy\) with limit values (0, 1) for x and y variables.

Solution:

Step 1: Compute The Inner Integral for variable x

 \( \ ∫_{0}^{1} (x^2 + 3xy^2 + xy) \, dx \)

\(\ = \left[ \frac{x^3}{3} + \frac{3}{2}x^2y^2 + \frac{x^2}{2}y \right]_{0}^{1} \)

\(\ = \left( \frac{1^3}{3} + \frac{3}{2}(1)y^2 + \frac{1^2}{2}y \right) - \left( \frac{0^3}{3} + \frac{3}{2}(0)y^2 + \frac{0^2}{2}y \right) \)

\(\ = \left( \frac{1}{3} + \frac{3}{2}y^2 + \frac{1}{2}y \right) - 0 \) \( = \frac{1}{3} + \frac{3}{2}y^2 + \frac{1}{2}y \)

Step 2: Now integrate the result obtained in step 1 for variable y

\(\ ∫_{0}^{1} \left( \frac{1}{3} + \frac{3}{2}y^2 + \frac{1}{2}y \right) \, dy \)

\(\ = \left[ \frac{1}{3}y + \frac{1}{2}y^3 + \frac{1}{4}y^2 \right]_{0}^{1} \)

\(\ = \left( \frac{1}{3}(1) + \frac{1}{2}(1)^3 + \frac{1}{4}(1)^2 \right) - \left( \frac{1}{3}(0) + \frac{1}{2}(0)^3 + \frac{1}{4}(0)^2 \right) \)

\(\ = \left( \frac{1}{3} + \frac{1}{2} + \frac{1}{4} \right) - 0 \)

\(\ = \frac{13}{12} \)

For triple integral calculations, check our Triple Integral Calculator.

How To Use The Double Integral Calculator?

• Enter a function that you want to integrate and select its integration order
• Choose the option whether you want to make definite or indefinite integration with this calculator and values accordingly to proceed with the integration method
• Click on the “Calculate Double Integral” button to evaluate the double integral step-wise

FAQ’s:

What Is double integrals used for?

• Finding Volume Under Surface: It is mostly used to find the volume under the surface
• Average Value: Double integrals are also used to compute the average value of the two-variable function f(x, y)
• Mass Distribution: They are used to calculate the mass of the materials having density ρ(x, y)
• Flux Calculation: Double integrals are used to compute the flux (rate of flow) of a vector field

What kind of functions can this calculator handle?

Our double integration calculator uses the double integration method to handle various types of 2-dimensional functions step-wise including:

• Polynomials
• Exponential Functions

Does order have any effect on double integration?

No, the order of integration can sometimes be switched while evaluating double integrals and this switch (change order of integration) can affect the difficulty of solving the integration. Still, it does not impact the double integration method.

According to Fubini's Theorem, if you have a two-variable function as f(x, y) which is continuous over a closed bounded region R, the double integral bounded by region computed for variable x first, then y (dxdy)is equal to the double integral performed for variable y first, then x (dydx). Here, the online calculator can help to define the double (multiple) integrals over the region ®.

What if I'm unsure about the bounds of integration?

The double integrals calculator can not find the bounds for you. You have to define them in the case of the double definite integral, otherwise use the indefinite integration for computing the given two-variable function without bounds.

Can I reverse the order of integration with this calculator?

Yes, the calculator for double integration can reverse the order of integration. It involves changing the order in which the integration is performed for two variables (x, y).

Is it possible to split the double integral?

Yes, double integrals are solved with iterated integration techniques (or repeated integration). It does not directly "split up" the double integral, but instead breaks it into two nested integrals, one for each integral. Also, the iterated integral calculator is a possible way to integrate a function of multiple variables.

Who can benefit from using this double integral calculator?

This calculator is very beneficial for students who are studying integration and professionals who are associated with engineering, physics, and science fields. Also, It is useful for people who have to perform double integrals upon the two variable functions as their preferences.

Reference:

From the source of Wikipedia: Multiple integrals, Methods of integration

From the source of libretexts.org: Double Integrals over General Regions