Find the average value of F(x, y, z) over the given region. F(x, y, z) = x2 + 9 over the cube in the first octant bounded by the coordinate planes and the planes x = 2, y = 2, and z =2.

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Find the average value of F(x, y, z) over the given region. F(x, y, z) = x2 + 9 over the cube in the first octant bounded by the coordinate planes and the planes x = 2, y = 2, and z =2.

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Answer 1

Let D be the volume of the cube bounded by the coordinate planes and the planes x = 2, y = 2, and z =2.
Then D = 2 * 2 * 2 = 8
Calculate average value of \(\displaystyle{x}^{{2}}+{9}\) over give cube D using formula:
Average value \(\displaystyle=\frac{{1}}{{{v}{o}{l}{u}{m}{e}\ {o}{f}\ {D}}}\int\int_{{D}}\int{F}{d}{V}\)
Volume of D is equal to \(\displaystyle{2}^{{3}}={8}\)
Average value \(\displaystyle=\frac{{1}}{{{v}{o}{l}{u}{m}{e}\ {o}{f}\ {D}}}{\int_{{0}}^{{2}}}{\int_{{0}}^{{2}}}{\int_{{0}}^{{2}}}{\left({x}^{{2}}+{9}\right)}{\left.{d}{z}\right.}{\left.{d}{y}\right.}{\left.{d}{x}\right.}\)
\(\displaystyle=\frac{{1}}{{8}}{\int_{{0}}^{{2}}}{\int_{{0}}^{{2}}}{\left({x}^{{2}}{z}+{9}{z}\right)}{{\mid}_{{0}}^{{2}}}{\left.{d}{y}\right.}{\left.{d}{x}\right.}\) (Integrate in relation to z.)
\(\displaystyle=\frac{{1}}{{8}}{\int_{{0}}^{{2}}}{\int_{{0}}^{{2}}}{\left({2}{x}^{{2}}+{18}\right)}{\left.{d}{y}\right.}{x}{e}\) (Compute the boundaries)
\(\displaystyle=\frac{{1}}{{8}}{\int_{{0}}^{{2}}}{\left({2}{x}^{{2}}{y}+{18}{y}\right)}{{\mid}_{{0}}^{{2}}}{\left.{d}{x}\right.}\) (Intergrate in relation to y)
\(\displaystyle=\frac{{1}}{{8}}{\int_{{0}}^{{2}}}{\left({4}{x}^{{2}}+{36}\right)}{\left.{d}{x}\right.}\) (Compute the boundaries)
\(\displaystyle=\frac{{1}}{{8}}{\left({4}\frac{{x}^{{3}}}{{3}}+{36}{x}\right)}{{\mid}_{{0}}^{{2}}}\) (Intergrate in relation to x)
\(\displaystyle=\frac{{1}}{{8}}{\left(\frac{{32}}{{3}}+{72}\right)}\) (Compute the boundaries)
\(\displaystyle=\frac{{4}}{{3}}+{9}=\frac{{31}}{{3}}\)
Final answer \(\displaystyle=\frac{{31}}{{3}}\)

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Find the average value of F(x, y, z) over the given region. F(x, y, z) = x2 + 9 over the cube in the first octant bounded by the coordinate planes and the planes x = 2, y = 2, and z =2.

Find the average value of F(x, y, z) over the given region. F(x, y, z) = x2 + 9 over the cube in the first octant bounded by the coordinate planes and the planes x = 2, y = 2, and z =2.

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