00:01
Hi, in this question we are given with a solid e that lies under the plane z is equals to 3 plus x plus y and above the region in xy plane that is bounded by the curve y is equals to under root x, y is equal to 0 and x is equal to 1 and it is given as the density function for this region is given to be 8.
00:30
So to start with the mass as we know m mass can be calculated as triple integral as the integration for x will be from 0 to 1, for y will be from 0 to under root x and for z will be from 0 to 3 plus x plus y here we have rho which is 8 times dz dy dx and solving this further we get this to be equals to integration from 0 to 1, integration from 0 to under root x here 8 can be taken outside integrating with respect to z we get 3 plus x plus y times dy dx.
01:18
Now solving this further we get as we will integrate now with respect to y we get 8 times from 0 to 1 here we get 3y plus xy plus y square by 2 and the limits from 0 to under root x dx substituting the limits we get 8 times integration from 0 to 1, 3 under root x plus x times under root x plus x by 2 dx and integrating this further we get this as equals to 8 times 3 times x raise to power 3 by 2 divide by 3 by 2 plus x raise to power 5 by 2 divide by 5 by 2 plus 1 by 2 times x square by 2 and the limits from 0 to 1.
02:08
Substituting the limits and solving further we get a final result for m that it will be equals to 106 by 5.
02:17
Next we will solve for the center of mass and for that we will first find the value for m yz where it will be equals to integration from 0 to 1, integration from 0 to under root x, integration from 0 to 3 plus x plus y times here we get 8x dz dy dx where this will be further equals to 8 times integration from 0 to 1, integration from 0 to under root x here integrating with respect to z we get x times 3 plus x plus y dy dx.
02:57
Now we will integrate with respect to y and we get 8 times integration from 0 to 1 here we get 3xy plus x square y plus y square by 2 limits from 0 to under root x here we get dx and substituting the limits and solving further we get this as equals to 8 times integration from 0 to 1, 3x root x plus x square root x plus x here we have y as we can see here we have x as well over here so we have x square and dx.
03:43
Solving this further we get a final result for m yz as it will be equals to 1388 by 105.
03:53
Now we will solve for m xz where it will be equals to integration from 0 to 1, integration from 0 to root x, integration from 0 to 3 plus x plus y here we get 8 times y times dz dy dx where this will be equals to 8 times integration from 0 to 1, integration from 0 to root x, y times 3 plus x plus y times dy dx and further solving we get this as 8 times integration from 0 to 1 here we get 3 times y square by 2 plus x times y square by 2 plus y raised to power 3 by 3 limits from 0 to root x dx and solving this further after substituting the limit we get the final result that will be equals to 42 by 5...