Double Integral Calculator: Get Step-by-Step Solutions!

Double Integral Calculator

Enter the function f(x, y) to calculate the double integral (antiderivative) with this calculator.

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Double Integral Calculator:

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

What Is Double Integral?

Double integral is an integration method used to find the volume under a surface, compute areas, mass, and flux over a 2D region. It integrates a function of two variables f(x, y) over a given area (denoted by R).

➥ Notation:

∫∫ _Rf (x, y) dA

➥ Expanded Form:

∫∫_R f (x, y) dxdy

∫∫_R f (x, y) dydx

Where:

∫∫ = double integral
R = Region of integration in the xy-plane
f(x, y) = Function being integrated
dA = Area element which can be written as dxdy or dydx

How To Solve Double Integrals?

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

Define the Region R: Identify the limits for x and y
Set Up the Integral: Write the double integral in the form: ∫∫_R f(x, y) dA
Evaluate the Inner Integral: Integrate with respect to one variable while treating the other as a constant
Evaluate the Outer Integral: Integrate the result with respect to the second variable
Obtain the Final Result: This gives the double integral over the region R

Example:

Evaluate double integral x² + 3xy² + 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?

Input the integrand f (x, y)
Fill in the upper and lower bounds
Select dxdy or dydx based on your problem
Click "Calculate"

- Click the camera icon to upload an equation image or the keyboard icon for a virtual input.

- Press the "Load Example" button for a sample calculation (Optional)

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 the order of integration matter?

No, the order can be changed without affecting the result. According to Fubini’s Theorem, if f(x, y), if f(x, y) is continuous over a region R, then:

∫∫_R f (x, y) dxdy = ∫∫_R f (x, y) dydx

What if I don’t know 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?

Yes, this double integral solver allows switching the order of integration between dxdy and dydx.

Who Can Benefit From Using This Double Integral Calculator?

This calculator is very beneficial for students studying integration and professionals associated with engineering, physics, and science fields. It is also useful for people who have to perform double integrals on two-variable functions according to their preferences.

References:

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

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