An integral over 3 variables.

∭_{E} f (x) d V

Cartesian

∭_{E} f (x, y, z) d x d y d z

where $x$ is the amount a point lies in the positive $x$ direction where $y$ is the amount a point lies in the positive $y$ direction where $z$ is the amount a point lies in the positive $z$ direction

Cylindrical

Polar Coordinates + $z$

∭_{E} f (r, θ, z) \cdot r d r d θ d z

where the Jacobian Determinant $det J_{f} = r$
where $r$ is the distance of a point’s $x y$ Projection from the origin
where $θ$ is the angle of the point from the positive $x$ axis
where $z$ is the amount a point lies in the positive $z$ direction
where $x = r cos θ, y = r sin θ, z = z$
where $r^{2} = x^{2} + y^{2}$
where $θ \in [0, 2 π], r \geq 0$

Spherical

∭_{E} f (ρ, θ, ϕ) \cdot ρ^{2} sin (ϕ) d r d θ d z

where the Jacobian Determinant $det J_{f} = ρ^{2} sin (ϕ)$
where $ρ$ is the distance of a point from the origin
where $θ$ is the angle of the point from the positive $x$ axis
where $ϕ$ is the angle of the point from the positive $z$ axis
where $r = ρ sin ϕ$
where $x = ρ sin ϕ cos θ, y = ρ sin ϕ sin θ, z = ρ cos ϕ$
where $ρ^{2} = r^{2} + z^{2}$
where $θ \in [0, 2 π], ϕ \in [0, π], ρ \geq 0$

Notes

Explorer

Triple Integral

Cartesian

Cylindrical

Spherical

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