Double and triple integrals
Accumulate a density or function across a region or solid.
Double and triple integrals is one of 3 multivariable formulas in the calculus section of this library, and it is used at university level.
Why double and triple integrals works
A single integral adds the function times a small length. A double integral adds it times a small patch of area, and a triple integral times a small block of space. The idea never changes: chop the region into tiny pieces, weight each piece by the function, and total them up.
What each symbol means
$R$ is a planar region, $E$ a spatial region, and $dA,dV$ are area and volume elements.
Double and triple integrals: when it holds
Choose bounds and Jacobian appropriate to the coordinate system; the integrals must converge.
When it stops applying
Changing coordinate systems without the scaling factor breaks it. In polar coordinates a small patch of area is the radius times the two small steps, not just the steps, and leaving that radius out gives a wrong value with the wrong units.
Double and triple integrals: a worked example
$\iint_{[0,1]^2}(x+y)dA=1$.
The mistake to avoid
What people do: Keeping the same limits after swapping the order of integration.
Why it goes wrong: The inner limits usually depend on the outer variable, so copying them across unchanged describes a rectangular box instead of the actual region.
Do this instead: Sketch the region, then read the boundaries off again in the new order before writing anything down.
Double and triple integrals: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $R$ is a planar region, $E$ a spatial region, and $dA,dV$ are area and volume elements.
- Check the conditions before substituting. Choose bounds and Jacobian appropriate to the coordinate system; the integrals must converge.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Multivariable
- Level
- University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where double and triple integrals comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Derivative Rules — the lesson behind this formula: differentiate sums, products, quotients, and compositions.
- Triple Integral Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about double and triple integrals
Does the order of integration change the answer?
For a continuous function on a reasonable region the value is the same, but one order can be dramatically easier, sometimes the difference between solvable and not.
What does integrating the constant 1 give?
Over a flat region it returns the area of that region, and over a solid it returns the volume, since every tiny piece contributes just its own size.
When should I switch to polar or spherical coordinates?
When the region is a disk, a ring, a ball, or a cone. Circular coordinates turn awkward square-root boundaries into constant limits.
Is this really just two ordinary integrals?
It is evaluated as nested ordinary integrals, but the inner one runs while the outer variable is held fixed, which is what lets the inner limits depend on it.