Double Integral Calculator with Graph
'/%3E%3Ccircle cx='48' cy='39' r='19' fill='%23f2c9a5'/%3E%3Cpath d='M27 35c3-16 39-17 42 2-8-8-27-9-42-2z' fill='%23374151'/%3E%3Cpath d='M21 86c5-24 49-28 56 0z' fill='%232563eb'/%3E%3Ccircle cx='41' cy='41' r='2' fill='%23111827'/%3E%3Ccircle cx='55' cy='41' r='2' fill='%23111827'/%3E%3Cpath d='M42 52c4 3 9 3 13 0' stroke='%2392400e' stroke-width='3' fill='none' stroke-linecap='round'/%3E%3C/svg%3E)
Reviewed by Calculator Editorial Team
Double integrals are used to calculate quantities like volume under a surface, mass of a region, or average value over a region. This calculator computes double integrals and displays the result graphically.
What is a Double Integral?
A double integral extends the concept of a single integral to two dimensions. It calculates the volume under a surface defined by a function z = f(x,y) over a region R 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:
- f(x,y) is the integrand function
- R is the region of integration
- dA is the differential area element
- g1(x) and g2(x) define the lower and upper bounds for y
- a and b define the bounds for x
How to Use This Calculator
- Enter the function f(x,y) you want to integrate
- Specify the region of integration by entering the bounds for x and y
- Click "Calculate" to compute the double integral
- View the result and the graphical representation
For best results, use simple functions and standard regions. Complex functions may require numerical methods.
Formula
The double integral is calculated using the formula:
∫∫R f(x,y) dA = ∫ab ∫g1(x)g2(x) f(x,y) dy dx
Where:
- f(x,y) is the function to integrate
- R is the region of integration
- a and b are the lower and upper bounds for x
- g1(x) and g2(x) are the lower and upper bounds for y
Example Calculation
Let's calculate the double integral of f(x,y) = x² + y² over the region R defined by 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.
∫01 ∫01 (x² + y²) dy dx
The result is approximately 1.3333.
You can verify this calculation using our calculator by entering the function and bounds.
FAQ
What is the difference between single and double integrals?
A single integral calculates area under a curve in one dimension, while a double integral calculates volume under a surface in two dimensions.
When would I use a double integral?
Double integrals are used in physics, engineering, and mathematics to calculate quantities like volume, mass, and average values over a region.
What types of functions can be integrated?
Most continuous functions can be integrated. However, some functions may require special techniques or numerical methods.