26.4. EXERCISES 531
26.4 Exercises1. Find the following iterated integrals.
(a)∫ 3−1∫ 2z
0∫ z+1
y (x+ y)dxdydz
(b)∫ 1
0∫ z
0∫ z2
y (y+ z)dxdydz
(c)∫ 3
0∫ x
1∫ 3x−y
2 sin(x)dzdydx
(d)∫ 1
0∫ 2x
x∫ 2y
y dzdydx
(e)∫ 4
2∫ 2x
2∫ x
2y dzdydx
(f)∫ 3
0∫ 2−5x
0∫ 2−x−2y
0 2x dzdydx
(g)∫ 2
0∫ 1−3x
0∫ 3−3x−2y
0 x dzdydx
(h)∫
π
0∫ 3y
0∫ y+z
0 cos(x+ y) dxdzdy
(i)∫
π
0∫ 4y
0∫ y+z
0 sin(x+ y) dxdzdy
2. Fill in the missing limits.∫ 10∫ z
0∫ z
0 f (x,y,z) dxdydz =∫ ?
?∫ ?
?∫ ?
? f (x,y,z) dxdzdy,∫ 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.