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Double Integral Calculator

Integration MethodNumerical Midpoint RuleSimpson's Rule (Approximation)

Function f(x, y):

x Lower Limit (a):

x Upper Limit (b):

y Lower Limit (c):

y Upper Limit (d):

Sub-intervals (n):

Show integration boundaries in result

Volume / Area Result:

∫∫ f(x,y) dA = —

How to Use the Double Integral Calculator

This double integral calculator is a powerful tool designed to solve multivariable calculus problems involving volume under a surface or the area of a region. By performing numerical integration, it approximates the value of a function $f(x, y)$ over a specified rectangular boundary $[a, b] \times [c, d]$.

To use the tool effectively, follow these steps:

Function f(x, y): Enter the expression you wish to integrate. Use standard operators like * for multiplication, ** or ^ for exponents, and Math.sin(x) for trigonometric functions.
x and y Limits: Define the rectangular region by setting the lower and upper bounds for both variables. The calculator assumes the order $dy dx$ or $dx dy$ over a fixed rectangle.
Sub-intervals (n): This determines the precision. A higher number of intervals (e.g., 100) creates a finer grid, leading to a more accurate approximation of the volume.

How It Works: The Math Behind Double Integrals

A double integral represents the volume between the xy-plane and a surface $z = f(x, y)$. When you use this double integral calculator, it utilizes the Riemann Sum (Midpoint Rule) to estimate the value. The fundamental formula for a double integral over a rectangle $R$ is:

$$\iint_R f(x,y) \, dA = \int_a^b \int_c^d f(x,y) \, dy \, dx$$

The numerical process breaks the region into $n \times n$ small rectangles, each with area $\Delta A = \Delta x \Delta y$. The calculator then sums the function values at the center of each small rectangle:

Step 1: Divide the interval $[a, b]$ into $n$ parts ($\Delta x$).
Step 2: Divide the interval $[c, d]$ into $n$ parts ($\Delta y$).
Step 3: Calculate the midpoint of each sub-rectangle.
Step 4: Evaluate $f(x, y)$ at each midpoint and multiply by the sub-area $\Delta A$.
Step 5: Sum all values to find the total approximated volume.

Calculation Example

Example: Calculate the volume under the surface $f(x, y) = x + y$ over the region $0 \le x \le 1$ and $0 \le y \le 1$.

Step-by-step solution:

Set $f(x, y) = x + y$
Set $x$ limits: $a = 0, b = 1$
Set $y$ limits: $c = 0, d = 1$
Analytical Solution: $\int_0^1 \int_0^1 (x+y) dy dx = \int_0^1 [xy + \frac{y^2}{2}]_0^1 dx = \int_0^1 (x + 0.5) dx$
Final Step: $[ \frac{x^2}{2} + 0.5x ]_0^1 = 0.5 + 0.5 = 1.0$
Calculator Result: If you enter these values into the tool with 50 intervals, the result will be exactly 1.000000.

Common Questions

What is the difference between a single and double integral?

A single integral calculates the area under a curve in 2D space. A double integral calculates the volume under a surface in 3D space, integrating across two different variables (usually x and y).

Can this calculator handle non-rectangular regions?

This specific double integral calculator is designed for rectangular regions where the limits of $y$ are constants. For regions where the limits of $y$ are functions of $x$ (Type I or Type II regions), you would typically need symbolic CAS software, though you can often transform those regions into rectangles using coordinate changes (like Polar coordinates).

Why is the result slightly different from my textbook?

Because this is a numerical calculator, it uses approximations. Increasing the "Sub-intervals" value will reduce the error and make the result closer to the exact analytical answer found in textbooks.