3.8 Fubini’s Theorem for Triple Integrals
Theorem 3.8.1 (Fubini’s theorem for triple integrals). If \(f\) is continuous on the rectangular box \(B = [a,b]\times [c,d] \times [r,s]\), then \[\iiint \limits _B f(x,y,z)\,dV = \int ^s_r\int ^d_c\int ^b_a f(x,y,z)\,dx \,dy \,dz ,\] and the iterated integral may be taken in any of the six possible orders.
Example 3.8.2. Evaluate \(\iiint _B xyz^2\, dV\), where \(B = \{ (x,y,z) \mid 0\leq x\leq 1,\ -1 \leq y \leq 2,\ 0\leq z\leq 3\}\).
Solution. The integrand is a product of functions of one variable each, so the integral factors: \[\iiint \limits _B xyz^2\, dV = \int ^1_0 x\,dx\int ^2_{-1}y\,dy\int ^3_0 z^2\,dz = \frac 12\cdot \frac 32\cdot 9 = \frac {27}{4} .\]
Example 3.8.3. Evaluate \(\iiint _B (x + y)e^{z}\,dV\) over \(B = [0,1]\times [0,2]\times [0,\ln 2]\).
Solution. Integrating \(z\) first, \(\int _0^{\ln 2}e^z\,dz = 1\), so the integral is \(\int _0^1\int _0^2 (x + y)\,dy\,dx = \int _0^1 (2x + 2)\,dx = 3\).
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