Key takeaways
- A double integral adds f(x, y) over a flat region: ∬ f dA.
- Over a rectangle, do the inner integral first, then the outer one.
- If f = g(x)·h(y), the answer is (∫g dx) × (∫h dy).
- If f is a height, the double integral is the volume under the surface.
In this guide
What is a double integral?
A normal integral adds up thin strips under a curve to get an area. A double integral adds up tiny squares over a flat region of the xy-plane. On each small square with area dA, you multiply by the value f(x, y) there, and then add everything.
A good picture: a rectangle of land with a roof above it at height f(x, y). The double integral is the volume of air between the ground and the roof. If f is a density, like kilograms per square metre, the double integral gives the total mass instead.
Why do double integrals matter?
The real world is not one-dimensional. Rain falls over a whole field, heat spreads across a whole floor, and a metal plate can be thicker in some places than others. Double integrals add up quantities like these over an area.
The good news is that you never need a new technique. A double integral is just two ordinary integrals, one inside the other.
Integrate f = 1 over a region and you get its area. Integrate a height and you get a volume. Integrate a density and you get a mass. Same tool, three meanings.
The formula
| Symbol | What it means | Example |
|---|---|---|
| R | The rectangle a ≤ x ≤ b, c ≤ y ≤ d | [0, 2] × [0, 3] |
| f(x, y) | What you add up (height, density…). Here xⁿ·yᵐ | x·y² |
| a, b | The x-limits | 0 and 2 |
| c, d | The y-limits | 0 and 3 |
| dA | A tiny piece of area, dx × dy | — |
This calculator uses f(x, y) = xⁿ · yᵐ. Because this splits into an x-part times a y-part, the double integral becomes a simple product: (∫ₐᵇ xⁿ dx) × (∫ᶜᵈ yᵐ dy). It needs a < b and c < d.
How to evaluate a double integral step by step
- 1Do the inner integral first. Integrate in x, treating y as a constant.
- 2Put in the x-limits. Substitute b and a and subtract. You now have an expression in y only.
- 3Do the outer integral. Integrate that result in y from c to d.
- 4Use the shortcut when you can. For a product like xⁿyᵐ, just multiply two separate definite integrals.
Now try your own powers and limits. The calculator below shows each of the two integrals and their product.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Mixing up the limits. The x-limits go with dx and the y-limits go with dy.
- Forgetting to treat the other variable as a constant. In the inner x-integral, y behaves like a plain number.
- Using fixed limits for curved regions. For triangles or circles, the inner limits depend on the other variable.
- Forgetting units. Density in kg/m² over an area in m² gives kg.
Tips and tricks
- Average value: divide the double integral by the area of the region. For the example,
18 ÷ 6 = 3kg per m². - Quick check: set f = 1 and you should get the area
(b − a)(d − c). - Changing variables? You'll need the Jacobian to rescale dA.
Where you'll use it in real life
- Engineering: the total mass of a thin plate with uneven thickness.
- Architecture: the volume under a curved roof or dome.
- Weather: total rainfall over a region from rainfall per square metre.
- Statistics: probabilities for two random variables at once.
Quick summary
A double integral adds f(x, y) over a flat region. Over a rectangle, integrate in x, then in y; and if f = xⁿ·yᵐ, just multiply the two single integrals. Use the calculator above to see both integrals and the final answer.
Worked example
A metal plate covers the rectangle 0 ≤ x ≤ 2, 0 ≤ y ≤ 3 (in metres). Its density is x·y² kilograms per square metre, so it gets heavier towards one corner. What is its total mass? Use n = 1, m = 2, a = 0, b = 2, c = 0 and d = 3.
- 01∬_R x^1·y^2 dA over [0,2]×[0,3]
- 02= (∫[0,2] x^1 dx) × (∫[0,3] y^2 dy) [separable]
- 03∫[0,2] x^1 dx = [x^2/2]|[0,2] = 2
- 04∫[0,3] y^2 dy = [y^3/3]|[0,3] = 9
- 05∬ = 2 × 9 = 18
These numbers are pre-filled in the calculator above.
Frequently asked questions
Does the order of integration matter in a double integral?
For a continuous function over a rectangle, no. You can integrate x first or y first and get the same answer (this is called Fubini's theorem).
What does a double integral represent?
If f(x, y) is positive, it is the volume under the surface z = f(x, y) above the region. It can also mean total mass, charge or probability, depending on what f measures.
What is the double integral of xy over the unit square?
It is (∫₀¹ x dx) × (∫₀¹ y dy) = (1/2) × (1/2) = 1/4.
Can this calculator handle circles or triangles?
No. It works over rectangles with f = xⁿ·yᵐ. For other regions the inner limits depend on the outer variable, so set them up by hand.
