Which integral runs first?

The inner integral runs first, so its bounds may depend on the outer variable. The outer bounds must be numeric or constant.

Does this page show a derivation?

It reports the parsed expression and result, but it does not claim textbook step-by-step working.

Can it integrate over a region that is not a rectangle?

Yes. Put the outer variable in the inner bounds: with the integrand 1, x from 0 to y and y from 0 to 3 describe a triangle, and the page gives 9/2, its area.

Is a double integral a volume?

Where the integrand is not negative over the region, yes: it is the volume under the surface z = f(x, y). x^2 + y^2 with x and y both from 0 to 1 gives 2/3, the volume under that bowl over the unit square.

What about polar coordinates or a third variable?

For polar coordinates, use r and t as the variables and multiply the integrand by r, so r alone over r from 0 to 1 and t from 0 to 2*pi gives pi. For a third variable with its own bounds, switch Integrals to Triple.

Double Integral Calculator

Switches the integral calculator to two dimensions with separate variables and bounds. The inner integral runs first, so its bounds may use the outer variable, which is how a triangle or a disc is described, and a definite value the sampling of the region contradicts is not shown.

IntegralWorking∬ x² dx dy · read as x^2 · symbolic

Integrand

Method

Integrals

Result: Working
Inner variable
Inner bounds
Numeric: n/a

Outer variable

Outer bounds

The inner integral runs first, so its bounds may use the outer variable and the outer bounds may not use the inner one.

x^2 dxindefinite
x³/3 + C
x^2 from 0 to 3definite
9
exp(x^2) dxno elementary form
named, not approximated

Accuracy. Exact symbolic integration for the functions it supports, with the expression it actually read printed back so you can check the parse. An integral with no elementary antiderivative is named as such rather than approximated silently, and no step-by-step derivation is claimed.

What you can type

Numbers, single letters for variables, the constants pi and e, the signs + - * / ^ and brackets, and these functions: sin, cos, tan, sec, csc, cot, asin, acos, atan, sinh, cosh, tanh, exp, sqrt, abs, log. ln is read as log, which is the natural logarithm here, and arcsin, arccos and arctan are read as asin, acos and atan. Multiplication can be left out where it is obvious, so 2x, 3 sin(x) and 2(x + 1) all read the way you would write them on paper, and the stage prints back what that came to. Leave both bounds empty for an antiderivative, or fill them in for a definite integral. Powers go up to 64, and the expression can be up to 500 characters long.

Common questions

Which integral runs first?: The inner integral runs first, so its bounds may depend on the outer variable. The outer bounds must be numeric or constant.
Does this page show a derivation?: It reports the parsed expression and result, but it does not claim textbook step-by-step working.
Can it integrate over a region that is not a rectangle?: Yes. Put the outer variable in the inner bounds: with the integrand 1, x from 0 to y and y from 0 to 3 describe a triangle, and the page gives 9/2, its area.
Is a double integral a volume?: Where the integrand is not negative over the region, yes: it is the volume under the surface z = f(x, y). x^2 + y^2 with x and y both from 0 to 1 gives 2/3, the volume under that bowl over the unit square.
What about polar coordinates or a third variable?: For polar coordinates, use r and t as the variables and multiply the integrand by r, so r alone over r from 0 to 1 and t from 0 to 2*pi gives pi. For a third variable with its own bounds, switch Integrals to Triple.

Exact symbolic integration for the functions it supports, with the expression it actually read printed back so you can check the parse. An integral with no elementary antiderivative is named as such rather than approximated silently, and no step-by-step derivation is claimed.