3.8 Fubini’s Theorem for Triple Integrals
If \(f\) is continuous on a rectangular box \([a,b]\times [c,d] \times [r,s]\) then \[\iiint \limits _B f(x,y,z)\hspace {0.1cm} = \int ^s_r\int ^d_c\int ^b_a f(x,y,z)\hspace {0.1cm}dx \hspace {0.1cm}dy \hspace {0.1cm}dz\]
Evaluate the triple integral \[\iiint \limits _B xyz^2\hspace {0.1cm} dx\hspace {0.1cm} dy\hspace {0.1cm}dz \] where \(B\) is given as \[B = \big \{ (x,y,z)\hspace {0.1cm} \big |\hspace {0.1cm} 0\leq x\leq 1\hspace {0.1cm} , \hspace {0.1cm} -1 \leq y \leq 2 \hspace {0.1cm}, \hspace {0.1cm} 0\leq z\leq 3\big \}\]
Solution. \begin {align*} \iiint \limits _B xyz^2\hspace {0.1cm} dx\hspace {0.1cm} dy\hspace {0.1cm}dz & = \int ^3_0 \int ^2_{-1}\int ^1_0 xyz^2\hspace {0.1cm} dx\hspace {0.1cm} dy\hspace {0.1cm}dz\\ & = \frac {27}{4}\\\\ \end {align*}
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