Triple Integral: Step-by-Step Guide
A triple integral integrates a function f(x, y, z) over a 3D region E and is evaluated as three nested single integrals. Triple integrals compute volumes, masses, centers of mass, and moments of inertia, with the integrand chosen to match the physical quantity.
Math Racer skill guide · Calculus 3 · College
How to solve it, step by step
- Describe the region E. Express E as constant outer bounds with successively dependent inner bounds, or use cylindrical or spherical coordinates if appropriate.\[E = \{(x,y,z) : a\le x\le b,\; g_1(x)\le y\le g_2(x),\; h_1(x,y)\le z\le h_2(x,y)\}\]
- Choose an order of integration. Pick an order (e.g., dz dy dx) so that the innermost variable's limits are easiest to describe in terms of the others.\[\iiint_E f\,dV = \int_a^b\!\!\int_{g_1}^{g_2}\!\!\int_{h_1}^{h_2} f\,dz\,dy\,dx\]
- Evaluate the innermost integral. Integrate over z (or chosen innermost variable), treating the others as constants.\[\int_{h_1}^{h_2} f(x,y,z)\,dz = F_1(x,y)\]
- Evaluate the middle integral. Integrate F_1(x, y) over y with limits g_1(x) to g_2(x).\[\int_{g_1(x)}^{g_2(x)} F_1(x,y)\,dy = F_2(x)\]
- Evaluate the outermost integral. Integrate F_2(x) from a to b. The result is the value of the triple integral.\[\int_a^b F_2(x)\,dx\]
Worked example
Evaluate the triple integral of xyz over the box E = [0,1] x [0,2] x [0,3].
Step 1: E is a rectangular box, so all bounds are constants.
Step 2: Set up as integral from x=0 to 1 of integral from y=0 to 2 of integral from z=0 to 3 of xyz dz dy dx.
Step 3: Innermost integral over z: integral from 0 to 3 of xyz dz = xy * z^2/2 from 0 to 3 = (9/2) xy.
Step 4: Middle integral over y: integral from 0 to 2 of (9/2) xy dy = (9/2) x * y^2/2 from 0 to 2 = (9/2) x * 2 = 9x.
Step 5: Outer integral over x: integral from 0 to 1 of 9x dx = 9 * x^2/2 from 0 to 1 = 9/2.
Verification: Since the integrand factors and the region is a box, the triple integral equals the product of single integrals: (integral 0 to 1 of x dx)(integral 0 to 2 of y dy)(integral 0 to 3 of z dz) = (1/2)(2)(9/2) = 9/2. Matches.
Tips and common mistakes
- For separable integrands over a box, the triple integral factors into a product of three 1D integrals.
- Always describe the region carefully: outer bounds constants, inner bounds depend on outer variables.
- Use cylindrical coordinates (r, theta, z) for regions with circular symmetry; dV = r dr d(theta) dz.
- Use spherical coordinates (rho, phi, theta) for spherical regions; dV = rho^2 sin(phi) drho dphi dtheta.
- If integrand or region is awkward in one order, switch to a different order of integration.
- Sketching cross-sections is essential to set up complicated 3D regions.
- Triple integral of 1 over E gives the volume of E.
- For mass, the integrand is density rho(x,y,z); for center of mass moments include x, y, or z factors.