Double Integral Calculator

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Double Integral Calculator
Rectangular Region (Cartesian: dx dy)
Result:

∫∫ f(x,y) dA = —

function calculateIntegral(){var funcStr=document.getElementById('functionInput').value;var xmin=parseFloat(document.getElementById('x_min').value);var xmax=parseFloat(document.getElementById('x_max').value);var ymin=parseFloat(document.getElementById('y_min').value);var ymax=parseFloat(document.getElementById('y_max').value);if(isNaN(xmin)||isNaN(xmax)||isNaN(ymin)||isNaN(ymax)){alert('Please enter valid numerical limits.');return;}var cleanFunc=funcStr.replace(/sin/g,'Math.sin').replace(/cos/g,'Math.cos').replace(/tan/g,'Math.tan').replace(/pow/g,'Math.pow').replace(/sqrt/g,'Math.sqrt').replace(/exp/g,'Math.exp').replace(/log/g,'Math.log').replace(/PI/g,'Math.PI');var f;try{f=new Function('x','y','return '+cleanFunc);}catch(e){alert('Invalid function syntax. Use standard JS math like x*x + y*y');return;}var steps=100;var dx=(xmax-xmin)/steps;var dy=(ymax-ymin)/steps;var totalSum=0;for(var i=0;i<steps;i++){var x=xmin+(i+0.5)*dx;for(var j=0;j<steps;j++){var y=ymin+(j+0.5)*dy;try{totalSum+=f(x,y);}catch(e){alert('Error evaluating function at x='+x+', y='+y);return;}}}var result=totalSum*dx*dy;document.getElementById('resultValue').innerHTML=result.toFixed(6);}

How to Use the Double Integral Calculator

The double integral calculator is a powerful tool designed for students, engineers, and mathematicians to evaluate the volume under a surface within a defined rectangular region. By entering a multivariable function $f(x,y)$ and specifying the integration limits for both $x$ and $y$, you can find the iterated integral quickly and accurately.

To use this calculator, follow these steps:

Function f(x,y)
Enter the expression you wish to integrate. Use standard mathematical operators: * for multiplication, / for division, and standard JS functions like sin(x), cos(y), or sqrt(x).
x Limits (a to b)
The lower and upper boundaries for the outer or inner integral with respect to the x-axis.
y Limits (c to d)
The lower and upper boundaries for the outer or inner integral with respect to the y-axis.

How It Works: The Math Behind Double Integrals

A double integral extends the concept of a single definite integral to functions of two variables. While a single integral represents the area under a curve, a double integral represents the volume under a surface $z = f(x, y)$ over a region $R$ in the xy-plane. The general formula for a rectangular region is:

V = ∫ (from c to d) ∫ (from a to b) f(x, y) dx dy

The calculation process involves:

  • Inner Integration: Integrating $f(x,y)$ with respect to $x$ while treating $y$ as a constant.
  • Evaluation: Applying the boundaries $a$ and $b$ to the resulting antiderivative.
  • Outer Integration: Integrating the resulting single-variable function of $y$ with respect to $y$.
  • Final Evaluation: Applying boundaries $c$ and $d$ to find the final numerical value.

Calculation Example

Example: Evaluate the double integral of $f(x,y) = x^2 + y$ over the region $0 \le x \le 2$ and $1 \le y \le 3$.

Step-by-step solution:

  1. Identify limits: $a=0, b=2, c=1, d=3$.
  2. Inner integral: $\int_0^2 (x^2 + y) dx = [x^3/3 + xy]_0^2 = (8/3 + 2y) – (0)$.
  3. Outer integral: $\int_1^3 (8/3 + 2y) dy$.
  4. Evaluate outer: $[8y/3 + y^2]_1^3 = (8(3)/3 + 3^2) – (8(1)/3 + 1^2) = (8 + 9) – (8/3 + 1) = 17 – 11/3$.
  5. Result: $51/3 – 11/3 = 40/3 \approx 13.3333$.

Common Questions

Does the order of integration (dx dy vs dy dx) matter?

According to Fubini's Theorem, if the function $f(x,y)$ is continuous on the rectangular region, the order of integration does not change the final result. However, for non-rectangular regions (Type I or Type II), changing the order requires changing the limits of integration.

What if the result is negative?

A negative double integral means that the "signed volume" below the xy-plane is greater than the volume above it. If you are calculating physical volume, you must take the absolute value of the regions accordingly.

How accurate is this numerical calculator?

This calculator uses a Midpoint Riemann Sum with 10,000 subdivisions (100×100). For standard polynomial and trigonometric functions, this provides a result accurate to approximately 4-6 decimal places.

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