Integral Calculator 2 Variables
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Reviewed by Calculator Editorial Team
This integral calculator solves double integrals with two variables. It handles both definite and indefinite integrals, supports various integration orders, and provides visualizations of the integration region.
What is a Double Integral?
A double integral extends the concept of single integration to two dimensions. It calculates the volume under a surface defined by a function of two variables, f(x,y), over a region in the xy-plane.
The double integral is expressed as:
∫∫R f(x,y) dA = ∫ab ∫g1(x)g2(x) f(x,y) dy dx
Where R is the region of integration, and dA represents an infinitesimal area element.
How to Use This Calculator
- Enter the integrand function f(x,y) in the first field
- Specify the limits of integration for x and y
- Select the integration order (dxdy or dydx)
- Click "Calculate" to compute the integral
- Review the result and visualization
For definite integrals, enter numerical limits. For indefinite integrals, leave the limit fields empty.
The Double Integral Formula
The general formula for a double integral in rectangular coordinates is:
∫∫R f(x,y) dA = ∫ab [∫g1(x)g2(x) f(x,y) dy] dx
For polar coordinates, the formula becomes:
∫∫R f(r,θ) r dr dθ
Worked Example
Let's calculate the volume under the surface z = x² + y² over the region where 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.
- Set up the integral: ∫01 ∫01 (x² + y²) dy dx
- First integrate with respect to y: ∫01 (x²y + y³/3) from 0 to 1 = x² + 1/12
- Then integrate with respect to x: ∫01 (x² + 1/12) dx = 1/3 + 1/12 = 5/12
The volume is 5/12 cubic units.
FAQ
- What is the difference between dxdy and dydx integration orders?
- The order of integration determines the limits of integration. Switching the order requires changing the limits accordingly.
- Can this calculator handle triple integrals?
- No, this calculator is specifically designed for double integrals with two variables.
- What if my function has singularities in the region?
- The calculator will attempt to compute the integral but may return an error if the integral doesn't converge.
- How accurate are the results?
- The calculator uses numerical methods for complex integrals, so results are accurate to about 6 decimal places.
- Can I use polar coordinates with this calculator?
- Yes, select "Polar" in the coordinate system dropdown to use polar coordinates.