What functions can I enter?

Any expression using x and y as variables with standard JavaScript Math operators: polynomials (x^2 + y^2), trig (sin(x)*cos(y)), exponentials (exp(x+y) or e^x), logarithms (log(x*y) or ln(x)), absolute value (abs(x-y)), and more. Use * for multiplication explicitly (2*x, not 2x).

How accurate is the result?

The midpoint Riemann sum with n=100 (10,000 sub-rectangles) gives accuracy within roughly 0.01% for smooth functions over moderate domains. Increasing n to 200 improves accuracy ~4× but takes longer. For highly oscillatory or singular functions, accuracy decreases.

What does the grid size n control?

n is the number of equal sub-intervals along each axis, so the total number of sub-rectangles evaluated is n². A grid of n=50 evaluates 2,500 points; n=200 evaluates 40,000 points. Larger n reduces step error but increases computation time.

Why does my result show NaN or Infinity?

NaN means the function returned an undefined value at one or more grid points (e.g. log of a negative number, 0/0, or a syntax error). Check that your function syntax is valid and that the domain [a,b]×[c,d] avoids singularities.

Can I integrate over non-rectangular regions?

This tool supports rectangular regions only. To integrate over a non-rectangular region, define f(x,y) = 0 outside the desired region using a conditional expression, or split the region into rectangles and sum the results.

Double Integral Numeric

Name: Double Integral Numeric
Availability: InStock
Author: Nham Vu

Enter f(x,y), set x and y bounds, and get a numeric approximation of the double integral using a 2D Riemann sum.

Configure Integral

f(x, y)

Use x and y as variables. Use * for multiply.

x bounds [a, b]

a (x lower)

b (x upper)

y bounds [c, d]

c (y lower)

d (y upper)

Grid size n (sub-intervals per axis)

Examples

Set the function and bounds, then click Calculate.

About the method

The tool divides the rectangle [a,b]×[c,d] into n×n sub-rectangles of equal size. For each sub-rectangle, it evaluates f at the midpoint and multiplies by the area Δx·Δy. The total is the sum of all those products — the midpoint Riemann sum. For smooth functions, error decreases as O(h²) where h = max(Δx, Δy), so doubling n roughly quarters the error.

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Summary

Enter f(x,y), set x and y bounds, and get a numeric approximation of the double integral using a 2D Riemann sum.

How it works

Enter the function f(x,y) using standard notation (e.g. x*y, sin(x+y), x^2 + y^2).
Set the x lower bound (a) and x upper bound (b).
Set the y lower bound (c) and y upper bound (d).
Choose the grid size n (number of sub-intervals per axis; higher n = more accurate).
Click "Calculate" to run the 2D midpoint Riemann sum.
The numeric result appears instantly with the step size and grid count used.

Use cases

Verify double integral results from calculus homework or textbooks.
Estimate volume under a surface when an antiderivative is difficult to find.
Compute 2D probability integrals for joint density functions.
Explore how integration bounds affect the integral value interactively.
Check symbolic CAS output against a numeric approximation.
Demonstrate iterated integration concepts in a classroom setting.

Frequently Asked Questions

Last updated: 2026-07-23 · Reviewed by Nham Vu