Question

Question:Write five other iterated integrals that are equal to the given iterated integral.

\(\int_{\rm{0}}^{\rm{1}} {\int_{\rm{y}}^{\rm{1}} {\int_{\rm{0}}^{\rm{z}} {\rm{f}} } } {\rm{(x,y,z)dx\;dz\;dy}}\)

Short answer

Hint: Make use of projections \({\rm{E}}\)onto planes \({\rm{xy,yz}}\)and \({\rm{xz}}\).

Step-by-step solution

Concept Introduction

A multiple integral is a definite integral of a function of many real variables in mathematics (particularly multivariable calculus), such as f(x, y) or f(x, y) (x, y, z). Integrals of a two-variable function over a region

Explanation of the solution

1) We begin by defining \({\rm{E}}\) the expression we have been given, resulting in:\(E = \{ (x,y,z)\mid 0 \le x \le z,y \le z \le 1,0 \le y \le 1\} \)

2) It's now critical to remember that \({\rm{E}}\) is restricted from below by two different surfaces, so keep that in mind while you design your projection onto the \({\rm{xy}}\)plane

3) Drawing of a projection \({\rm{E}}\)onto \({\rm{xy}}\)plane:

4) Drawing of a projection \({\rm{E}}\)onto \({\rm{xy}}\)plane:

5) Notice that the projection onto the \({\rm{yz}}\) plane is the same as the projection onto the \({\rm{xz}}\)plane:

6) The equivalent integrals can now be written.

\(\begin{array}{c}\int_{\rm{0}}^{\rm{1}} {\int_{\rm{y}}^{\rm{1}} {\int_{\rm{0}}^{\rm{z}} {\rm{f}} } } {\rm{(x,y,z)dx\;dz\;dy = }}\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{\rm{x}} {\int_{\rm{x}}^{\rm{1}} {\rm{f}} } } {\rm{(x,y,z)dz\;dy\;dx + }}\int_{\rm{0}}^{\rm{1}} {\int_{\rm{x}}^{\rm{1}} {\int_{\rm{y}}^{\rm{1}} {\rm{f}} } } {\rm{(x,y,z)dz\;dy\;dx}}\\{\rm{ = }}\int_{\rm{0}}^{\rm{1}} {\int_{\rm{y}}^{\rm{1}} {\int_{\rm{x}}^{\rm{1}} {\rm{f}} } } {\rm{(x,y,z)dz\;dx\;dy + }}\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{\rm{y}} {\int_{\rm{y}}^{\rm{1}} {\rm{f}} } } {\rm{(x,y,z)dz\;dx\;dy}}\\{\rm{ = }}\int_{\rm{0}}^{\rm{1}} {\int_{\rm{x}}^{\rm{1}} {\int_{\rm{0}}^{\rm{z}} {\rm{f}} } } {\rm{(x,y,z)dy\;dz\;dx}}\\{\rm{ = }}\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{\rm{z}} {\int_{\rm{0}}^{\rm{z}} {\rm{f}} } } {\rm{(x,y,z)dy\;dx\;dz}}\\{\rm{ = }}\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{\rm{z}} {\int_{\rm{0}}^{\rm{z}} {\rm{f}} } } {\rm{(x,y,z)dx\;dy\;dz}}\end{array}\)

Make use of projections \({\rm{E}}\)onto planes \({\rm{xy,yz}}\)and \({\rm{xz}}\).