Kenneth Kuttler

360 CHAPTER 19. THE RIEMANNN INTEGRAL ON Rn

∫ 10∫ z

0∫ 2z

0 f (x,y,z) dxdydz =∫ ?

?∫ ?

? f (x,y,z) dydzdx,∫ 10∫ z

0∫ z

0 f (x,y,z) dxdydz =∫ ?

?∫ ?

? f (x,y,z) dzdydx,∫ 10∫√z

z/2

∫ y+z0 f (x,y,z) dxdydz =

∫ ??∫ ?

?∫ ?

? f (x,y,z) dxdzdy,∫ 64∫ 6

2∫ 4

0 f (x,y,z) dxdydz =∫ ?

?∫ ?

? f (x,y,z) dzdydx.

3. Find the volume of R where R is the bounded region formed by the plane 15 x+ y+

14 z = 1 and the planes x = 0,y = 0,z = 0.

4. Find the volume of R where R is the bounded region formed by the plane 15 x+ 1

2 y+14 z = 1 and the planes x = 0,y = 0,z = 0.

5. Find the volume of R where R is the bounded region formed by the plane 15 x+ 1

3 y+14 z = 1 and the planes x = 0,y = 0,z = 0.

6. Find the volume of the bounded region determined by 3y+ z = 3,x = 4− y2,y =0,x = 0.

7. Find the volume of the region bounded by x2 + y2 = 16,z = 3x,z = 0, and x≥ 0.

8. Find the volume of R where R is the bounded region formed by the plane 14 x+ 1

2 y+14 z = 1 and the planes x = 0,y = 0,z = 0.

9. Here is an iterated integral:∫ 3

0∫ 3−x

0∫ x2

0 dzdydx. Write as an iterated integral in thefollowing orders: dzdxdy, dxdzdy, dxdydz, dydxdz, dydzdx.

10. Find the volume of the bounded region determined by 2y+ z = 3,x = 9− y2,y =0,x = 0,z = 0.

11. Find the volume of the bounded region determined by y+ 2z = 3,x = 9− y2,y =0,x = 0.

12. Find the volume of the bounded region determined by y+z = 2,x = 3−y2,y = 0,x =0.

13. Find the volume of the region bounded by x2 + y2 = 25,z = x,z = 0, and x≥ 0.

Your answer should be 2503 .

14. Find the volume of the region bounded by x2 + y2 = 9,z = 3x,z = 0, and x≥ 0.

19.4.1 Mass And DensityAs an example of the use of triple integrals, consider a solid occupying a set of pointsU ⊆ R3 having density ρ . Thus ρ is a function of position and the total mass of the solidequals

∫U ρ dV . This is just like the two dimensional case. The mass of an infinitesimal

chunk of the solid located at x would be ρ (x) dV and so the total mass is just the sum ofall these,

∫U ρ (x) dV .

Example 19.4.1 Find the volume of R where R is the bounded region formed by the plane15 x+ y+ 1

5 z = 1 and the planes x = 0,y = 0,z = 0.

36010.11.12.13.14.CHAPTER 19. THE RIEMANNN INTEGRAL ON R"Jo Jo Jo’ f (952) dxdydz = fy fy fz f (&.y,2) dydzdx,Jo IG Io f (9,2) dxdydz = fy fy fy f (x,y,z) dedydx,Jo Lvs Jaf (%5y,2) dxdydz = fy fy fy f (%y,2) dxdzdy,I? J Jo f 09,2) dxdydz = fy fy’ Sy f (x,y,z) dzdydx.Find the volume of R where R is the bounded region formed by the plane 5x +y+42 = | and the planes x = 0,y = 0,z=0.Find the volume of R where R is the bounded region formed by the plane 5x + sy +iz = | and the planes x = 0,y = 0,z=0.Find the volume of R where R is the bounded region formed by the plane gx+ 5y +42 = | and the planes x = 0,y = 0,z=0.Find the volume of the bounded region determined by 3y+ z= 3,x =4—y*,y =0,x=0.Find the volume of the region bounded by x” + y? = 16,z = 3x,z = 0, and x > 0.Find the volume of R where R is the bounded region formed by the plane gat sy +qz= 1 and the planes x = 0,y = 0,z=0.Here is an iterated integral: fo ° i i dzdydx. Write as an iterated integral in thefollowing orders: dzdxdy, dxdzdy, dxdy dz, dydxdz, dydzdx.Find the volume of the bounded region determined by 2y +z = 3,x =9—y*,y =0,x =0,z=0.Find the volume of the bounded region determined by y+ 2z = 3,x =9-—y*,y=0,x=0.Find the volume of the bounded region determined by y+ z= 2,x =3 — y’, y=0,x=0.Find the volume of the region bounded by x* + y* = 25,z =x,z = 0, and x > 0.Your answer should be 250Find the volume of the region bounded by x* + y* = 9,z = 3x,z = 0, and x > 0.19.4.1 Mass And DensityAs an example of the use of triple integrals, consider a solid occupying a set of pointsU CR?» having density p. Thus p is a function of position and the total mass of the solidequals f,; dV. This is just like the two dimensional case. The mass of an infinitesimalchunk of the solid located at a would be p (a) dV and so the total mass is just the sum ofall these, fi, p (a) dV.Example 19.4.1 Find the volume of R where R is the bounded region formed by the planeax+yt+ $2 = | and the planes x =0,y = 0,z=0.