19.5. EXERCISES 363

8. Find the mass of the bounded region R formed by the plane 14 x+ 1

2 y+ 13 z = 1 and the

planes x = 0,y = 0,z = 0 if the density is ρ (x,y,z) = y

9. Find the mass of the bounded region R formed by the plane 12 x+ 1

2 y+ 14 z = 1 and the

planes x = 0,y = 0,z = 0 if the density is ρ (x,y,z) = z2

10. Find the mass of the bounded region R formed by the plane 14 x+ 1

2 y+ 14 z = 1 and the

planes x = 0,y = 0,z = 0 if the density is ρ (x,y,z) = y+ z

11. Find the mass of the bounded region R formed by the plane 14 x+ 1

2 y+ 15 z = 1 and the

planes x = 0,y = 0,z = 0 if the density is ρ (x,y,z) = y

12. Find∫ 1

0∫ 12−4z

0∫ 3−z

14 y

sinxx dxdydz.

13. Find∫ 20

0∫ 2

0∫ 6−z

15 y

sinxx dxdzdy+

∫ 3020∫ 6− 1

5 y0

∫ 6−z15 y

sinxx dxdzdy.

14. Find the volume of the bounded region determined by x ≥ 0,y ≥ 0,z ≥ 0, and 12 x+

y+ 12 z = 1, and x+ 1

2 y+ 12 z = 1.

15. Find the volume of the bounded region determined by x ≥ 0,y ≥ 0,z ≥ 0, and 17 x+

y+ 13 z = 1, and x+ 1

7 y+ 13 z = 1.

16. Find an iterated integral for the volume of the region between the graphs of z =x2 + y2 and z = 2(x+ y).

17. Find the volume of the region which lies between z = x2 + y2 and the plane z = 4.

18. The base of a solid is the region in the xy plane between the curves y = x2 and y = 1.The top of the solid is the plane z = 2− x. Find the volume of the solid.

19. The base of a solid is in the xy plane and is bounded by the lines y = x,y = 1−x, andy = 0. The top of the solid is z = 3− y. Find its volume.

20. The base of a solid is in the xy plane and is bounded by the lines x = 0,x = π,y = 0,and y = sinx. The top of this solid is z = x. Find the volume of this solid.