CalculusFree

Double Integral Calculator

Evaluate double integrals of polynomial functions over rectangular regions. Enter the integrand as a*x^m*y^n with bounds for both variables.

Enter Values

Coefficient (a)

The constant multiplier in f(x,y) = a*x^m*y^n

x Exponent (m)

Power of x in the integrand

y Exponent (n)

Power of y in the integrand

x Lower Bound

x Upper Bound

y Lower Bound

y Upper Bound

Result

Enter values above and click Calculate to see your result.

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Formula

Core Formula

Integrate inner: F(y) = integral[xL to xU] f(x,y) dxIntegrate outer: integral[yL to yU] F(y) dyResult = Volume under f(x,y) over region

How it works: For a polynomial integrand f(x,y) = a*x^m*y^n over a rectangular region, integrate with respect to x first (treating y as constant), then integrate the result with respect to y. The final value represents the volume under the surface.

Worked Example

Evaluate double integral of 2x*y over [0,3] x [0,2]:

1Step 1: Inner integral (x): integral from 0 to 3 of 2xy dx = x^2*y from 0 to 3 = 9y

2Step 2: Outer integral (y): integral from 0 to 2 of 9y dy = 9y^2/2 from 0 to 2 = 18

Result: 18

What Is a Double Integral?

A double integral is an extension of the definite integral concept to functions of two variables. While a single integral calculates the area under a curve in two dimensions, a double integral computes the volume under a three-dimensional surface defined by a function f(x,y) over a specified two-dimensional region in the xy-plane. When the region of integration is a rectangle, as supported by this calculator, the evaluation process simplifies to iterated integration. This means you perform two successive single integrations: first with respect to one variable, treating the other as a constant, and then with respect to the second variable. The result represents the signed volume between the surface and the xy-plane over the given region. Double integrals are fundamental in fields like physics and engineering for calculating quantities such as mass, moments of inertia, and centroids.

Double integrals measure the volume beneath a surface in three-dimensional space.
They are evaluated using iterated integration, integrating with respect to one variable at a time.
Crucial for problems involving functions of two independent variables.
Applied in physics, engineering, and statistics for calculating various spatial properties.
This calculator focuses on polynomial integrands over well-defined rectangular regions.

Mastering double integrals is key to understanding multivariable calculus concepts. Utilize this calculator to quickly solve problems and reinforce your knowledge of volumes and surface interactions.

You can also calculate changes using our Triple Integral Calculator or Partial Derivative Calculator.

Frequently Asked Questions

What does a double integral compute?

A double integral computes the volume under a surface f(x,y) over a two-dimensional region. For constant f, it gives the area of the region times the height.

What is a rectangular region?

A rectangular region is defined by constant bounds: x ranges from a to b and y ranges from c to d, independently. This is the simplest case for double integration.

Does the order of integration matter?

For continuous functions over rectangular regions, the order of integration does not affect the result (Fubini's theorem). For non-rectangular regions, the bounds may change form when swapping order.

How do I add this Double Integral Calculator to my site?

Absolutely. Use the "Embed" option above to tailor the dimensions, color scheme, and styling to match your site. Copy the generated iframe snippet and drop it into your HTML, WordPress editor, or any CMS. There is no cost and no account required. See calculory.com/services/embed-calculators for a step-by-step walkthrough.

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