Does the page add r² sin φ for me?

No. It integrates exactly what is in the integrand box, which is why the page opens with r^2*sin(p) already there. To integrate a function over a ball, multiply it into that factor: r^2 times the volume element, written r^4*sin(p), over the same bounds gives 4*pi/5.

Why p and t instead of φ and θ?

Every variable here is one Latin letter, so p stands for φ and t for θ, and the stage prints dr dp dt back so you can see which is which. Other letters work too, as long as the three differ.

Which order are the integrals taken in?

r first, then p, then t, matching the inner, middle and outer boxes. Bounds on r may use p and t, bounds on p may use t, and the bounds on t take numbers. Changing p to run from 0 to pi/2 and r to 2 gives the upper half of a ball of radius 2, 16*pi/3.

Spherical Coordinates Integral Calculator

Opens a triple integral in spherical coordinates: r is the distance from the origin, p the angle φ down from the z axis and t the angle θ around it. It starts on the volume element r^2*sin(p) alone, with r from 0 to 1, p from 0 to pi and t from 0 to 2*pi, which comes to 4*pi/3, the volume of the unit ball.

IntegralWorking∭ r²*sin(p) dr dp dt, r from 0 to 1, p from 0 to pi, t from 0 to 2*pi · read as r^2*sin(p) · 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 r² sin φ for me?: No. It integrates exactly what is in the integrand box, which is why the page opens with r^2*sin(p) already there. To integrate a function over a ball, multiply it into that factor: r^2 times the volume element, written r^4*sin(p), over the same bounds gives 4*pi/5.
Why p and t instead of φ and θ?: Every variable here is one Latin letter, so p stands for φ and t for θ, and the stage prints dr dp dt back so you can see which is which. Other letters work too, as long as the three differ.
Which order are the integrals taken in?: r first, then p, then t, matching the inner, middle and outer boxes. Bounds on r may use p and t, bounds on p may use t, and the bounds on t take numbers. Changing p to run from 0 to pi/2 and r to 2 gives the upper half of a ball of radius 2, 16*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.