Answer
$\dfrac{(a+c)^5b-c^5b-a^5b}{20}$
Work Step by Step
Given: $ f(x, y, z)=(x+z)^3$
The iterated triple integral can be calculated as:
$\iiint_{\mathcal{B}} f(x,y,z)d V = \iiint_{\mathcal{B}} (x+z)^3 d V \\
=\int_{0}^{c} \int_{0}^{b} \int_{0}^{a} (x+z)^3 \ dx \ dy \ dz \\
= \int_{0}^{c} \int_{0}^{b} [\dfrac{(x+z)^4}{4}]_0^a \ dy dz \ dx\\=\int_{0}^{c} \int_0^b [\dfrac{(a+z)^4}{4}-\dfrac{z^4}{4}]_0^b \ dz \\=\int_0^c [\dfrac{(a+z)^4b}{4}-\dfrac{z^4b}{4}] \ dz \\=[\dfrac{(a+z)^5b}{20}-\dfrac{z^5b}{20}]_0^c \\=\dfrac{(a+c)^5b-c^5b-a^5b}{20}$
