Find the mass of the surface $S$ having the given mass density.
$S$ is part of the plane $x + 5 y + 9 z = 45$ in the first octant; the density at a point $P$ on $S$ is equal to the square of the distance between $P$ and the $x y$-plane.
A) 42
B) $\sqrt { 107 }$
C) 19
D) $\frac { 375 \sqrt { 107 } } { 4 }$

Question

Quizplus · Accepted Answer

To find the mass of the surface $S$, we first need to express the density function and the limits of integration. The density at a point $P$ on $S$ is given as the square of the distance from $P$ to the $xy$-plane. Since the distance from any point to the $xy$-plane is simply its $z$-coordinate, the density function is $\rho(x,y,z) = z^2$.The equation of the plane is $x + 5y + 9z = 45$. To find the mass, we integrate the density function over the surface. We can solve for $z$ to get $z = \frac{45 - x - 5y}{9}$. The surface is in the first octant, so $x$, $y$, and $z$ are all non-negative, and $x + 5y \leq 45$.To set up the integral, we convert the equation of the plane to express $z$ in terms of $x$ and $y$, and then integrate over the appropriate limits for $x$ and $y$. The mass $M$ is given by the double integral over the surface $S$ of the density function:$$M = \iint_S \rho(x,y,z) \, dS$$Since $dS = \sqrt{1 + \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2} \, dx \, dy$ and $\frac{\partial z}{\partial x} = -\frac{1}{9}$, $\frac{\partial z}{\partial y} = -\frac{5}{9}$, we find that $dS = \sqrt{1 + \left(-\frac{1}{9}\right)^2 + \left(-\frac{5}{9}\right)^2} \, dx \, dy = \sqrt{\frac{107}{81}} \, dx \, dy$.Therefore, the mass is:$$M = \int_0^{45} \int_0^{\frac{45-x}{5}} \left(\frac{45 - x - 5y}{9}\right)^2 \sqrt{\frac{107}{81}} \, dy \, dx$$After performing the integration, the result is:$$M = \frac{375\sqrt{107}}{4}$$This calculation involves integrating the square of the $z$-value (which represents the density) over the area of the plane in the first octant, taking into account the differential area element $dS$ that includes the gradient of the plane to account for its orientation in space.

Find the Mass of the Surface SSS Having the Given Mass Density

Find the Mass of the Surface $S$ Having the Given Mass Density