1 point the motion of a solid object can be analyzed by...

(1 point) The motion of a solid object can be analyzed by thinking of the mass as concentrated at a single point, the center of mass. If the object has density \rho (x, y, z) at the point (x, y, z) and occupies a region W, then the coordinates (x̅, y̅, z̅) of the center of mass are given by x̅=1/m∫_W x \rho d V y̅=1/m∫_W y \rho d V z̅=1/m∫_W z \rho d V where m=∫_W\rho d V is the total mass of the body. Consider a solid is bounded below by the square z=0,0 <= x <= 4,0 <= y <= 5 and above by the surface z=x+y+4. Let the density of the solid be 1 g / cm^3, with x, y, z measured in cm. Find each of the following:

Transcript

00:01 Hi, in this question we will be dealing with a solid that is bounded by a square z is equal to 0, x varies from 0 to 2, y is varying from 0 to 5 and above the surface z is equal to x plus y plus 5 and density is given to be equals to 1 gram per centimeter cube.

00:25 With x, y, z all are measured in centimeter we need to find the mass of the solid where as we know we can find the mass that will be equals to integration from 0 to 5 for y, integration from 0 to 2 for x, integration from 0 to x plus y plus 5 for z, dz, dx, dy.

00:48 Now solving this further as we have rho over here, substituting the value of rho and this will be equals to integration from 0 to 5, integration from 0 to 2, here we have rho as 1 and integrating with respect to z we get z limits from 0 to x plus y plus 5 dx, dy and this will be further equals to integration from 0 to 5, integration from 0 to 2, here we get x plus y plus 5 times dx, dy and this will be equals to integration from 0 to 5, here we get x square by 2 plus yx plus 5x limits from 0 to 2 then we will integrate with respect to y and this will be equals to integration from 0 to 5, here we get 2 plus 10 plus 2y dy and further solving by integrating with respect to y we get this to be equals to 60 plus 25 and further solving we get the final result for the mass of the solid as m to be equals to 85 gram.

02:04 Next we will solve for the value of x coordinate of the solid and this will be equals to 1 by m integration from 0 to 5, integration from 0 to 2, integration from 0 to x plus y plus 5, here we have x times dz dx dy and solving this further we get this to be equals to 1 by 85 integration from 0 to 5, integration from 0 to 2, here integrating with respect to z we get x times x plus y plus 5 as we have density as 1 only and dx dy and this will be equals to 1 by 85 times integration from 0 to 5 and here integrating this we get x raise to power 3 by 3 plus y times x square by 2 plus 5 times x square by 2 limits from 0 to 2 then integrating with respect to y.

03:03 Now substituting the limits and integrating with respect to y we get the final result for x coordinate and this will be equals to 53 by 51...

Question

Please give Ace some feedback