Question

Consider the solid bounded by y=x²+z² and y=6 with density δ(x,y,z)=65−y. Assume density is measured in kg/m³. (a) Draw a graph of the given solid and include it with your written work. the entire integral by hand, showing all your work and giving an exact answer. dydrdθ= kg (c) Determine the center of mass of the solid in rectangular coordinates. You may use technology to completely evaluate any integrals. Round your answers to 4 decimal places.

Answer 1

To find the center of mass of the described solid, one needs to set up and evaluate triple integrals in cylindrical coordinates, and then divide the moments by the total mass to get the center of mass in rectangular coordinates.

Step-by-step explanation:

The student is asking about determining the center of mass of a solid with a given density function, bounded by a paraboloid and a plane. The density function is given by δ(x,y,z) = 65 - y, where x, y, and z are the coordinates in three-dimensional space. To find the center of mass, one would need to calculate the three coordinates separately by using triple integrals.

To do this, the problem should be set up in cylindrical coordinates because of the symmetry around the y-axis. This involves integrating over r (radius), θ (angular coordinate), and y (vertical coordinate). The integrals would calculate the moments in each coordinate direction and the total mass, which are then used to find the center of mass by dividing the moments by the total mass.

The calculation involves setting up the integral bounds for r from 0 to √(6), θ from 0 to 2π, and y from r² to 6. Triple integral techniques and sometimes technology are used to evaluate these complex integrals. The answer will be in the form of rectangular coordinates (x, y, z), rounded to four decimal places.