Exercise · ZW-TRIPLE-INTEGRAL-010 · ★★★ Advanced
For a continuous function on a box, how many iterated orders of integration are there and how are their results related?
Choose one option.
A There are \(3\) orders and they give different results. B There are \(6\) orders and all give the same result. C There are \(6\) orders and they give different results. D There are \(9\) orders and all give the same result. Show answer B. There are \(6\) orders and all give the same result.← Back to the question
Explanation The three variables can be ordered in \(3!=6\) ways, and by Fubini's theorem all give the same triple integral when \(f\) is continuous on the box.
The triple integral of a function \(f\) over a solid \(E\) of space is the limit of the sums of the values of \(f\) times the volume \(\Delta V\) of ever finer cells covering \(E\).
\[\iiint_E f\,dV=\lim_{\max\Delta V_i\to0}\sum_i f(x_i^*,y_i^*,z_i^*)\,\Delta V_i\]
More practice problems: triple integral