Question

Question: The figure shows the region of integration for the integral \(\int_{\rm{0}}^{\rm{1}} {\int_{\sqrt {\rm{x}} }^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - y}}} {{\rm{f(x,y,z)dzdydx}}} } } \)Rewrite this integral as an equivalent iterated integral in the five other orders.

Short answer

The integrals are,

\(\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{{{\rm{x}}^{\rm{2}}}} {\int_{\rm{0}}^{{\rm{1 - y}}} {{\rm{f(x,y,z)dzdxdy}}} } } \)

\(\begin{array}{l}\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - }}\sqrt {\rm{x}} } {\int_{\sqrt {\rm{x}} }^{{\rm{1 - z}}} {{\rm{f(x,y,z)dydzdx}}} } } \\\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{{{\left( {{\rm{1 - z}}} \right)}^{\rm{2}}}} {\int_{\sqrt {\rm{x}} }^{{\rm{1 - z}}} {{\rm{f(x,y,z)dydzdx}}} } } \end{array}\)

\(\begin{array}{l}\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - z}}} {\int_{\rm{0}}^{{{\rm{y}}^{\rm{2}}}} {{\rm{f(x,y,z)dxdydz}}} } } \\\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - y}}} {\int_{\rm{0}}^{{{\rm{y}}^{\rm{2}}}} {{\rm{f(x,y,z)dxdzdy}}} } } \end{array}\)

Step-by-step solution

Define integral

In mathematics, an integral assigns numbers to functions to express displacement, area, volume, and other concepts arising from linking infinitesimal data.

Explanation on xy plane projection

Integration Region,

\({\rm{E = \{ (x,y,z)|0}} \le {\rm{x}} \le {\rm{1,}}\sqrt {\rm{x}} \le {\rm{y}} \le {\rm{1,0}} \le {\rm{z}} \le {\rm{1 - y\} }}\)

\({\rm{xy}}\) plane projection,

Then,

\(\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{{{\rm{x}}^{\rm{2}}}} {\int_{\rm{0}}^{{\rm{1 - y}}} {{\rm{f(x,y,z)dzdxdy}}} } } \)

Explanation on xz plane projection

\({\rm{xz}}\) plane projection,

Then,

\(\begin{array}{l}\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - }}\sqrt {\rm{x}} } {\int_{\sqrt {\rm{x}} }^{{\rm{1 - z}}} {{\rm{f(x,y,z)dydzdx}}} } } \\\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{{{\left( {{\rm{1 - z}}} \right)}^{\rm{2}}}} {\int_{\sqrt {\rm{x}} }^{{\rm{1 - z}}} {{\rm{f(x,y,z)dydzdx}}} } } \end{array}\)

Explanation on yz plane projection

\({\rm{yz}}\) plane projection,

Then,

\(\begin{array}{l}\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - z}}} {\int_{\rm{0}}^{{{\rm{y}}^{\rm{2}}}} {{\rm{f(x,y,z)dxdydz}}} } } \\\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - y}}} {\int_{\rm{0}}^{{{\rm{y}}^{\rm{2}}}} {{\rm{f(x,y,z)dxdzdy}}} } } \end{array}\)