Find the mass of the surface S having the given mass density. S is part of the plane in the first octant; the density at a point P on S is equal to the square of the distance between P and the xy-plane.
B)49
D)20

Question

Find the mass of the surface S having the given mass density. S is part of the plane  in the first octant; the density at a point P on S is equal to the square of the distance between P and the xy-plane.
B)49 
D)20

Quizplus · Accepted Answer

The surface S is part of the plane in the first octant, which means that its boundary is defined by the x-axis, y-axis, and z=0 plane. Therefore, the integral for finding the mass of S can be set up as:

$\int\int_S \rho dS = \int_0^1 \int_0^x \int_0^{\sqrt{x^2+y^2}} \rho dz dy dx$

where $\rho = (x^2+y^2)^2$ is the mass density at a point P on S.

Evaluating this integral will give us the mass of S:

$\int_0^1 \int_0^x \int_0^{\sqrt{x^2+y^2}} \rho dz dy dx = \frac{1}{10}$

Therefore, the correct answer is (C) 11eaa3e5_54ba_8720_9f8f_e70b7ac6a003_TB5971_11, which is equivalent to $\frac{1}{10}$.

Find the Mass of the Surface S Having the Given