Double Integral Calculator

How to Use This Calculator

1. Enter f(x,y) (e.g. x*y).
2. Set x bounds [x₁, x₂] and y bounds [y₁, y₂] for the rectangle.
3. The double integral is computed using 2D Simpson's rule.
4. Use the Polar tab for circular/annular regions (enter f(r,θ) with variables r and t).
5. The Professional tab computes mass and centroid with a separate density function.

Formula

Example

Frequently Asked Questions

• What is a double integral? ▼
  A double integral ∫∫_R f(x,y) dA gives the signed volume under the surface z=f(x,y) over a region R in the xy-plane. It extends the single integral to two dimensions.
• What is Fubini's theorem? ▼
  Fubini's theorem states that for continuous f on a rectangle, ∫∫ f dA = ∫∫ f dx dy = ∫∫ f dy dx — the order of integration can be reversed. The tool verifies this numerically.
• How is the double integral computed numerically? ▼
  A 2D version of Simpson's rule is applied: the x and y intervals are each divided into n subintervals and the 2D weighted sum is evaluated. With nx=ny=50 this gives good accuracy for smooth functions.
• What is the Jacobian for polar coordinates? ▼
  When changing to polar coordinates x=r·cos(θ), y=r·sin(θ), the area element becomes dA = r dr dθ. The extra factor r is the Jacobian.
• How do I compute mass with a density function? ▼
  Mass = ∫∫ ρ(x,y) dA, where ρ is the density. The Professional tab accepts a separate density function and also computes the centroid (x̄, ȳ).

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