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This triple integral calculator evaluates \(\iiint f(x,y,z)\,dV\) one layer at a time. You enter the function and three sets of limits. It shows the inner, middle, and outer integrals as separate steps, then gives the exact value and a decimal. A separate numerical method double-checks the result.
How to Use It
- Type the function of x, y, and z, for example
x^2 + y^2 + z. Type1to find a volume. - Enter limits for z (inner), y (middle), and x (outer). Inner limits may use x and y. Middle limits may use x. Outer limits must be numbers.
- Press Solve.
What Is a Triple Integral?
A triple integral adds up a quantity over a 3D solid. If \(f = 1\), the answer is the volume of the solid. If \(f\) is density, the answer is the mass.
We work from the inside out:
\[\int_a^b\!\int_{g_1(x)}^{g_2(x)}\!\int_{h_1(x,y)}^{h_2(x,y)} f(x,y,z)\,dz\,dy\,dx\]
Worked Examples
Example 1: \(\int_0^3\!\int_0^2\!\int_0^1 (x^2+y^2+z)\,dz\,dy\,dx\)
Inner (z): \(\int_0^1 (x^2+y^2+z)\,dz = x^2 + y^2 + \frac12\).
Middle (y): \(\int_0^2 \left(x^2 + y^2 + \frac12\right)dy = 2x^2 + \frac83 + 1 = 2x^2 + \frac{11}{3}\).
Outer (x): \(\int_0^3 \left(2x^2 + \frac{11}{3}\right)dx = 18 + 11 = 29\).
Example 2: Volume of a solid, \(\int_0^1\!\int_0^x\!\int_0^{x+y} 1\,dz\,dy\,dx\)
Inner: \(\int_0^{x+y} 1\,dz = x + y\).
Middle: \(\int_0^x (x+y)\,dy = x^2 + \frac{x^2}{2} = \frac{3x^2}{2}\).
Outer: \(\int_0^1 \frac{3x^2}{2}\,dx = \frac12\). The volume is \(\frac12\).
Example 3: \(\int_0^1\!\int_0^1\!\int_0^1 xyz\,dz\,dy\,dx\)
The function splits into \(x \cdot y \cdot z\), so the answer is \(\frac12\cdot\frac12\cdot\frac12 = \frac18\).
Cylindrical and Spherical Coordinates
Round solids are often easier in other coordinates. Convert first, then use this calculator:
| System | Substitution | Volume element |
|---|---|---|
| Cylindrical | \(x = r\cos\theta,\ y = r\sin\theta,\ z = z\) | \(dV = r\,dz\,dr\,d\theta\) |
| Spherical | \(x = \rho\sin\phi\cos\theta,\ y = \rho\sin\phi\sin\theta,\ z = \rho\cos\phi\) | \(dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta\) |
For example, the volume of a ball of radius 1 is \(\int_0^{2\pi}\!\int_0^{\pi}\!\int_0^1 \rho^2\sin\phi\,d\rho\,d\phi\,d\theta = \frac{4\pi}{3}\). Type it as z^2 sin(y) with z from 0 to 1, y from 0 to pi, and x from 0 to 2pi.
Common Mistakes
- Wrong order of limits. The innermost limits can use the two outer variables, never the other way around.
- Forgetting the extra r or ρ²sin φ when changing coordinates.
- Losing track of constants. In the inner integral, both outer variables act like numbers.
Learn the Math Behind the Triple Integral Calculator
A triple integral adds up a function over a solid region, for volume, mass or average value in three dimensions.
Frequently Asked Questions
What order does this calculator use?
It uses \(dz\,dy\,dx\): z first, then y, then x. Rename variables in your function if your problem uses another order.
Can I find the mass of a solid?
Yes. Type the density function as \(f(x,y,z)\) and enter the limits of the solid.
What if I only have two variables?
Use the double integral calculator.