Question

Question: Find the mass and center of mass of the solid \({\rm{S}}\)bounded by the paraboloid \({\rm{z = 4}}{{\rm{x}}^{\rm{2}}}{\rm{ + 4}}{{\rm{y}}^{\rm{2}}}\) and the plane \({\rm{z = a(a}} \succ {\rm{0)}}\)if \(S\)has constant density\(K\).

Short answer

The mass and center of mass of the solid is\({\rm{m = }}\frac{{{{\rm{a}}^{\rm{2}}}{\rm{K\pi }}}}{{\rm{8}}}{\rm{,}}\left( {{\rm{0,0,}}\frac{{{\rm{2a}}}}{{\rm{3}}}} \right)\).

Step-by-step solution

To find the mass.

Formula:

as well as the axis of mass

Using cylindrical coordinates, solve these integrals.

\(\left\{ {\begin{array}{*{20}{l}}{{\rm{x = rcos\theta }}}\\{{\rm{y = rsin\theta }}}\\{{\rm{z = z}}}\\{{\rm{dxdydz = rdzdrd\theta }}}\end{array}{\rm{.}}} \right.\)

Sketch the given solid.

Evaluate.

Take a look at the \(z\)-axis. The bounds of \(z\)are as follows:

\({\rm{4}}{{\rm{x}}^{\rm{2}}}{\rm{ + 4}}{{\rm{y}}^{\rm{2}}} \le {\rm{z}} \le {\rm{a}} \Rightarrow {\rm{4}}{{\rm{r}}^{\rm{2}}} \le {\rm{z}} \le {\rm{a}}\)

The following is the \(xy\)-plane region:

\(\begin{array}{c}{\rm{4}}{{\rm{x}}^{\rm{2}}}{\rm{ + 4}}{{\rm{y}}^{\rm{2}}}{\rm{ = a}}\\{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{ = }}{\left( {\frac{{\sqrt {\rm{a}} }}{{\rm{2}}}} \right)^{\rm{2}}}\\{{\rm{r}}^{\rm{2}}}{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{\theta + }}{{\rm{r}}^{\rm{2}}}{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{\theta = }}{\left( {\frac{{\sqrt {\rm{a}} }}{{\rm{2}}}} \right)^{\rm{2}}}\\{\rm{r = }}\frac{{\sqrt {\rm{a}} }}{{\rm{2}}}\end{array}\)

As a result, the area in the \(xy\)plane is a circle with a radius of\(\frac{{\sqrt {\rm{a}} }}{{\rm{2}}}\). As a result, the \({\rm{r}}\)and \({\rm{\theta }}\)limitations are.

\(\begin{array}{c}{\rm{0}} \le {\rm{r}} \le \frac{{\sqrt {\rm{a}} }}{{\rm{2}}}\\{\rm{0}} \le {\rm{\theta }} \le {\rm{2\pi }}\end{array}\)

Because the solid is symmetric around the \(z\)-axis,

\({\rm{\bar x = \bar y = 0}}{\rm{.}}\)

To find mass\(m\).

\({\rm{m = }}\int_{\rm{0}}^{{\rm{2\pi }}} {\int_{\rm{0}}^{\sqrt {\rm{a}} {\rm{/2}}} {\int_{{\rm{4}}{{\rm{r}}^{\rm{2}}}}^{\rm{a}} {\rm{K}} } } {\rm{rdzdrd\theta }}\)

Integrate in terms of\(z\):

Examine the limitations.

Step 6: