Does the page add the extra r?

No. It integrates what is in the integrand box, so convert your function and multiply it by r yourself. x^2 + y^2 is r^2 in polar coordinates, so type r^2 times r, written r^3, and over the unit disc the page gives pi/2.

How do I turn a region into polar bounds?

Describe it by radius and angle. A quarter disc of radius 2 in the first quadrant is r from 0 to 2 and t from 0 to pi/2, which gives pi. A radius that changes with the angle goes in the r bounds as a function of t, because r is integrated first: r from 0 to 1 + cos(t) and t from 0 to 2*pi is a cardioid, and the page gives 3*pi/2.

Every variable here is one Latin letter, so t stands for θ, and the stage prints dr dt back so you can check the order. Any two different letters work.

Polar Double Integral Calculator

Opens a double integral in polar coordinates: r is the distance from the origin and t the angle θ. It starts on the area element r alone, with r from 0 to 1 and t from 0 to 2*pi, which comes to pi, the area of the unit disc.

IntegralWorking∬ r dr dt, r from 0 to 1, t from 0 to 2*pi · read as r · 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

Does the page add the extra r?: No. It integrates what is in the integrand box, so convert your function and multiply it by r yourself. x^2 + y^2 is r^2 in polar coordinates, so type r^2 times r, written r^3, and over the unit disc the page gives pi/2.
How do I turn a region into polar bounds?: Describe it by radius and angle. A quarter disc of radius 2 in the first quadrant is r from 0 to 2 and t from 0 to pi/2, which gives pi. A radius that changes with the angle goes in the r bounds as a function of t, because r is integrated first: r from 0 to 1 + cos(t) and t from 0 to 2*pi is a cardioid, and the page gives 3*pi/2.
Why t instead of θ?: Every variable here is one Latin letter, so t stands for θ, and the stage prints dr dt back so you can check the order. Any two different letters work.

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.