Question

Question: Express the integralas an iterated integral in six different ways, where \({\rm{E}}\) is the solid bounded by the given surfaces.

\({\rm{x = 2,y = 2,z = 0,x + y - 2z = 2}}\)

\({\rm{y = }}{{\rm{x}}^{\rm{2}}}{\rm{,z = 0,y + 2z = 4}}\).

Short answer

The integrals are,

\(\begin{array}{l}\int_{\rm{0}}^{\rm{2}} {\int_{{\rm{2 - x}}}^{\rm{2}} {\int_{\rm{0}}^{{\rm{0}}{\rm{.5(x + y - 2)}}} {{\rm{dzdydx}}} } } \\\int_{\rm{0}}^{\rm{2}} {\int_{{\rm{2 - y}}}^{\rm{2}} {\int_{\rm{0}}^{{\rm{0}}{\rm{.5(x + y - 2)}}} {{\rm{dzdxdy}}} } } \end{array}\)

\(\begin{array}{l}\int_{\rm{0}}^{\rm{1}} {\int_{{\rm{2z}}}^{\rm{2}} {\int_{{\rm{2 - y + 2z}}}^{\rm{2}} {{\rm{dxdydz}}} } } \\\int_{\rm{0}}^{\rm{2}} {\int_{\rm{0}}^{{\rm{0}}{\rm{.5y}}} {\int_{{\rm{2 - y + 2z}}}^{\rm{2}} {{\rm{dxdzdy}}} } } \end{array}\)

\(\begin{array}{l}\int_{\rm{0}}^{\rm{2}} {\int_{\rm{0}}^{{\rm{0}}{\rm{.5x}}} {\int_{{\rm{2 - x + 2z}}}^{\rm{2}} {{\rm{dydzdx}}} } } \\\int_{\rm{0}}^{\rm{1}} {\int_{{\rm{2z}}}^{\rm{2}} {\int_{{\rm{2 - x + 2z}}}^{\rm{2}} {{\rm{dydxdz}}} } } \end{array}\)

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

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

\(\begin{array}{l}{\rm{x = 2}}\\{\rm{y = 2}}\end{array}\)

\({\rm{x + y = 2}}\)

There are two methods to represent a bounded region.

\({\rm{R = \{ (x,y):0}} \le {\rm{x}} \le {\rm{2,2 - x}} \le {\rm{y}} \le {\rm{2\} }}\)

And

\({\rm{R = \{ (x,y):0}} \le {\rm{y}} \le {\rm{2,2 - y}} \le {\rm{x}} \le {\rm{2\} }}\)

\({\rm{z = 0}}\)and\({\rm{z = }}\frac{{{\rm{x + y - 2}}}}{{\rm{2}}}\)are the boundaries of coordinate\({\rm{z}}\).

In region\({\rm{Rx + y}} \ge {\rm{2}}\)for\({\rm{(x,y)}}\),

As a result,\({\rm{0}} \le {\rm{z}} \le \frac{{{\rm{x + y - 2}}}}{{\rm{2}}}\), solid\({\rm{E}}\)is given by,

\({\rm{E = }}\left\{ {{\rm{(x,y,z):0}} \le {\rm{x}} \le {\rm{2,2 - x}} \le {\rm{y}} \le {\rm{2,0}} \le {\rm{z}} \le \frac{{{\rm{x + y - 2}}}}{{\rm{2}}}} \right\}\)

And

\({\rm{E = }}\left\{ {{\rm{(x,y,z):0}} \le {\rm{y}} \le {\rm{2,2 - y}} \le {\rm{x}} \le {\rm{2,0}} \le {\rm{z}} \le \frac{{{\rm{x + y - 2}}}}{{\rm{2}}}} \right\}\)

Hence, the integrals that correspond to each other are,

\(\begin{array}{l}\int_{\rm{0}}^{\rm{2}} {\int_{{\rm{2 - x}}}^{\rm{2}} {\int_{\rm{0}}^{{\rm{0}}{\rm{.5(x + y - 2)}}} {{\rm{dzdydx}}} } } \\\int_{\rm{0}}^{\rm{2}} {\int_{{\rm{2 - y}}}^{\rm{2}} {\int_{\rm{0}}^{{\rm{0}}{\rm{.5(x + y - 2)}}} {{\rm{dzdxdy}}} } } \end{array}\)

Explanation on x plane projection

Plane \({\rm{x = 2}}\) projection,

\(\begin{array}{l}{\rm{z = 0,y = 2}}\\{\rm{y - 2z = 0}}\end{array}\)

There are two methods to represent a bounded region.

\({\rm{R = \{ (y,z):0}} \le {\rm{z}} \le {\rm{1,2z}} \le {\rm{y}} \le {\rm{2\} }}\)

And

\({\rm{R = \{ (y,z):0}} \le {\rm{y}} \le {\rm{2,0}} \le {\rm{z}} \le {\rm{0}}{\rm{.5y\} }}\)

\({\rm{x = 2}}\)and\({\rm{x = 2 - y + 2z}}\)are the boundaries of coordinate\({\rm{x}}\).

In region\({\rm{Ry - 2z}} \ge {\rm{0}}\)for\({\rm{(y,z)}}\),

As a result,\({\rm{2 - y + 2z}} \le {\rm{x}} \le {\rm{2}}\), solid\({\rm{E}}\)is given by,

\({\rm{E = }}\left\{ {{\rm{(x,y,z):0}} \le {\rm{z}} \le {\rm{1,2z}} \le {\rm{y}} \le {\rm{2,2 - y + 2z}} \le {\rm{x}} \le {\rm{2}}} \right\}\)

And

\({\rm{E = }}\left\{ {{\rm{(x,y,z):0}} \le {\rm{y}} \le {\rm{2,0}} \le {\rm{z}} \le {\rm{0}}{\rm{.5y,2 - y + 2z}} \le {\rm{x}} \le {\rm{2}}} \right\}\)

Hence, the integrals that correspond to each other are,

\(\begin{array}{l}\int_{\rm{0}}^{\rm{1}} {\int_{{\rm{2z}}}^{\rm{2}} {\int_{{\rm{2 - y + 2z}}}^{\rm{2}} {{\rm{dxdydz}}} } } \\\int_{\rm{0}}^{\rm{2}} {\int_{\rm{0}}^{{\rm{0}}{\rm{.5y}}} {\int_{{\rm{2 - y + 2z}}}^{\rm{2}} {{\rm{dxdzdy}}} } } \end{array}\)

Explanation on y plane projection

Plane \({\rm{y = 2}}\) projection,

\(\begin{array}{l}{\rm{z = 0,x = 2}}\\{\rm{x - 2z = 0}}\end{array}\)

There are two methods to represent a bounded region.

\({\rm{R = \{ (x,z):0}} \le {\rm{x}} \le {\rm{2,0}} \le {\rm{z}} \le {\rm{0}}{\rm{.5x\} }}\)

And

\({\rm{R = \{ (x,z):0}} \le {\rm{z}} \le {\rm{1,2z}} \le {\rm{x}} \le {\rm{2\} }}\)

\({\rm{x = 2}}\)and\({\rm{x = 2 - y + 2z}}\)are the boundaries of coordinate\({\rm{x}}\).

In region\({\rm{Rx - 2z}} \ge {\rm{0}}\)for\({\rm{(x,z)}}\),

As a result,\({\rm{2 - x + 2z}} \le {\rm{y}} \le {\rm{2}}\), solid\({\rm{E}}\)is given by,

\({\rm{E = }}\left\{ {{\rm{(x,y,z):0}} \le {\rm{x}} \le {\rm{2,0}} \le {\rm{z}} \le {\rm{0}}{\rm{.5x,2 - x + 2z}} \le {\rm{y}} \le {\rm{2}}} \right\}\)

And

\({\rm{E = }}\left\{ {{\rm{(x,y,z):0}} \le {\rm{z}} \le {\rm{1,2z}} \le {\rm{x}} \le {\rm{2,2 - x + 2z}} \le {\rm{y}} \le {\rm{2}}} \right\}\)

Hence, the integrals that correspond to each other are,

\(\begin{array}{l}\int_{\rm{0}}^{\rm{2}} {\int_{\rm{0}}^{{\rm{0}}{\rm{.5x}}} {\int_{{\rm{2 - x + 2z}}}^{\rm{2}} {{\rm{dydzdx}}} } } \\\int_{\rm{0}}^{\rm{1}} {\int_{{\rm{2z}}}^{\rm{2}} {\int_{{\rm{2 - x + 2z}}}^{\rm{2}} {{\rm{dydxdz}}} } } \end{array}\)