Does the page add the factor r for me?

No. It integrates what is in the integrand box, so the r is already there when the page opens. Multiply your function into it: the integral of r^2 over the same cylinder is r^3, which gives pi.

How is this different from polar coordinates?

The r and t part is a polar double integral, and z is integrated as it is, so a triple integral in polar coordinates with a height is this page. For a flat region with no height, switch Integrals to Double and leave z out.

Can the radius depend on the height?

Yes, because r is integrated first: its bounds may use t and z. With r from 0 to z, t from 0 to 2*pi and z from 0 to 1, the integrand r describes a cone standing on its point, and the page gives pi/3.

Cylindrical Coordinates Integral Calculator

Opens a triple integral in cylindrical coordinates: r is the distance from the z axis, t the angle θ around it and z the height. It starts on the volume element r alone, with r from 0 to 1, t from 0 to 2*pi and z from 0 to 2, which comes to 2*pi, the volume of that cylinder.

IntegralWorking∭ r dr dt dz, r from 0 to 1, t from 0 to 2*pi, z from 0 to 2 · read as r · symbolic

Integrand

Method

Integrals

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

Middle variable

Middle bounds

Outer variable

Outer bounds

The inner integral runs first, then the middle, then the outer. Inner bounds may use the middle and outer variables, middle bounds may use the outer one, and the outer bounds take numbers only.

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 factor r for me?: No. It integrates what is in the integrand box, so the r is already there when the page opens. Multiply your function into it: the integral of r^2 over the same cylinder is r^3, which gives pi.
How is this different from polar coordinates?: The r and t part is a polar double integral, and z is integrated as it is, so a triple integral in polar coordinates with a height is this page. For a flat region with no height, switch Integrals to Double and leave z out.
Can the radius depend on the height?: Yes, because r is integrated first: its bounds may use t and z. With r from 0 to z, t from 0 to 2*pi and z from 0 to 1, the integrand r describes a cone standing on its point, and the page gives pi/3.

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.