TITLE: Triple Integral Proof
QUESTION [5 upvotes]: Our professor gave us a question to try by ourselves at home, saying that $a_1 \leq x \leq a_2,\;b_1 \leq y \leq b_2,\;c_1 \leq z \leq c_2$ how would you show $$\iiint \alpha\;dxdydz = \alpha(a_2-a_1)(b_2-b_1)(c_2-c_1)$$ I solved it using iterated integrals but he said to attempt it without it and use the definition of a triple integral. I've been stumped at how to approach this, how would I start off?

REPLY [3 votes]: There is no need for iterated integrals
\begin{eqnarray*}
\iiint \alpha\;dxdydz =  \alpha \int_{a_1}^{a_2} dx \int_{b_1}^{b_2} dy \int_{c_1}^{c_2} dz = \cdots
\end{eqnarray*}