Triple Integral Calculator with Steps (Variable Bounds)

Q: What does a triple integral represent?

With integrand 1 it is the volume of the solid region E. With a density \rho(x,y,z) it is the mass; dividing \iiint x\rho\,dV by the mass gives the centre of mass.

Q: How do I find the bounds of a triple integral?

Pick the innermost variable (often z) and find the surfaces below and above the solid: those are its bounds, as functions of x and y. Then project the solid onto the plane of the other two variables and set up a double integral over that shadow region.

Q: What is the volume of the tetrahedron x + y + z ≤ 1?

\int_0^1\int_0^{1-x}\int_0^{1-x-y} 1\,dz\,dy\,dx = \frac16. Load it from the examples to see the three layers.

Q: Can I use cylindrical or spherical coordinates?

Yes, by including the Jacobian yourself: cylindrical dV = r\,dz\,dr\,d\theta, spherical dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta. Enter the transformed integrand with x, y, z standing in for the new variables.

Inside out, one variable at a time

\iiint_E f\,dV = \int_a^b \int_{g_1(x)}^{g_2(x)} \int_{h_1(x,y)}^{h_2(x,y)} f(x,y,z)\,dz\,dy\,dx

The innermost integral runs from the bottom surface $z = h_1(x,y)$ to the top surface $z = h_2(x,y)$ , collapsing each vertical column to a number. What remains is a double integral over the solid’s shadow in the xy-plane.

Common uses

Volume: integrand 1.
Mass: integrand = density $\rho(x,y,z)$ .
Average value: $\frac{1}{V}\iiint_E f\,dV$ .

Questions students ask

How do you solve a triple integral?

Work from the inside out. Integrate the innermost variable with the other two held constant, plug in its bounds, then repeat for the middle and outer variables. Each layer removes one variable until a number is left.

What does a triple integral represent?

With integrand 1 it is the volume of the solid region $E$ . With a density $\rho(x,y,z)$ it is the mass; dividing $\iiint x\rho\,dV$ by the mass gives the centre of mass.

How do I find the bounds of a triple integral?

Pick the innermost variable (often $z$ ) and find the surfaces below and above the solid: those are its bounds, as functions of $x$ and $y$ . Then project the solid onto the plane of the other two variables and set up a double integral over that shadow region.

What is the volume of the tetrahedron x + y + z ≤ 1?

$\int_0^1\int_0^{1-x}\int_0^{1-x-y} 1\,dz\,dy\,dx = \frac16$ . Load it from the examples to see the three layers.

Can I use cylindrical or spherical coordinates?

Yes, by including the Jacobian yourself: cylindrical $dV = r\,dz\,dr\,d\theta$ , spherical $dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta$ . Enter the transformed integrand with $x, y, z$ standing in for the new variables.

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Inside out, one variable at a time

Common uses

Questions students ask