Question

Question: The figure shows the region of integration for the integral

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

Rewrite this integral as an equivalent iterated integral in the five other orders.

Short answer

For the solution, look at your work.

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

Simplify equivalent iterated integral in the five other orders.

Order 1:

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

Order 2:

\(\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - }}{{\rm{x}}^{\rm{2}}}} {\int_{\rm{0}}^{{\rm{1 - x}}} {\rm{f}} } } {\rm{(x,y,z)dydzdx = }}\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{\sqrt {{\rm{1 - z}}} } {\int_{\rm{0}}^{{\rm{1 - x}}} {\rm{f}} } } {\rm{(x,y,z)dydxdz}}\)

Order 3:

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

Order 4:

\(\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - }}{{\rm{x}}^{\rm{2}}}} {\int_{\rm{0}}^{{\rm{1 - x}}} {\rm{f}} } } {\rm{(x,y,z)dydzdx = }}\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{2y - }}{{\rm{y}}^{\rm{2}}}} {\int_{\rm{0}}^{{\rm{1 - y}}} {\rm{f}} } } {\rm{(x,y,z)dxdzdy + }}\int_{\rm{0}}^{\rm{1}} {\int_{{\rm{2y - }}{{\rm{y}}^{\rm{2}}}}^{\rm{1}} {\int_{\rm{0}}^{\sqrt {{\rm{1 - y}}} } {\rm{f}} } } {\rm{(x,y,z)dxdzdy}}\)

Order 5:

\(\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - }}{{\rm{x}}^{\rm{2}}}} {\int_{\rm{0}}^{{\rm{1 - x}}} {\rm{f}} } } {\rm{(x,y,z)dydzdx = }}\int_{\rm{0}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - }}\sqrt {{\rm{1 - z}}} } {\int_{\rm{0}}^{\sqrt {{\rm{1 - z}}} } {\rm{f}} } } {\rm{(x,y,z)dxdydz + }}\int_{\rm{0}}^{\rm{1}} {\int_{{\rm{1 - }}\sqrt {{\rm{1 - z}}} }^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - y}}} {\rm{f}} } } {\rm{(x,y,z)dxdydz}}\)