Let W be the region below the paraboloid x2 + y2 = z 2 that lies above the part of the

Which of (a)(c) is not equal to 1 0 4 3 7 6 f (x, y, z) dz dy dx? (a) 7 6 1 0 4 3 f (x, y, z) dy dx dz (b) 4 3 1 0 7 6 f (x, y, z) dz dx dy (c) 1 0 4 3 7 6 f (x, y, z) dx dz dy 2. Which of

Which of the following is not a meaningful triple integral? (a) 1 0 x 0 2x+y x+y ex+y+z dz dy dx (b) 1 0 z 0 2x+y x+y ex+y+z dz dy dx 3. De

Describe the projection of the region of integration W onto the xy-plane: (a) 1 0 x 0 x2+y2 0 f (x, y, z) dz dy dx (b) 1 0 1x2 0 4 2 f (x, y, z) dz dy dx Exerci

In Exercises 18, evaluate B f (x, y, z) dV for the specified function f and box B. 1.f (x, y, z) = x (y + z)2 ; [0, 2][2, 4] [1, 1]

In Exercises 18, evaluate B f (x, y, z) dV for the specified function f and box B. 1.f (x, y, z) = (x y)(y z); [0, 1][0, 3][0, 3]

In Exercises 18, evaluate B f (x, y, z) dV for the specified function f and box B. 1.f (x, y, z) = z x ; 1 x 3, 0 y 2, 0 z 4

In Exercises 18, evaluate B f (x, y, z) dV for the specified function f and box B. 1.f (x, y, z) = (x + z)3; [0, a][0, b][0, c]

In Exercises 18, evaluate B f (x, y, z) dV for the specified function f and box B. 1.f (x, y, z) = (x + y z)2; [0, a][0, b][0, c]

In Exercises 914, evaluate W f (x, y, z) dV for the function f and region W specified. 9. f (x, y, z) = x + y; W : y z x, 0 y x, 0 x 1 10

In Exercises 914, evaluate W f (x, y, z) dV for the function f and region W specified. 9.f (x, y, z) = ex+y+z; W : 0 z 1, 0 y x, 0 x 1

In Exercises 914, evaluate W f (x, y, z) dV for the function f and region W specified. 9.f (x, y, z) = xyz; W : 0 z 1, 0 y  1 x2, 0 x 1

In Exercises 914, evaluate W f (x, y, z) dV for the function f and region W specified. 9.f (x, y, z) = x; W : x2 + y2 z 4

In Exercises 914, evaluate W f (x, y, z) dV for the function f and region W specified. 9.f (x, y, z) = ez; W : x + y + z 1, x 0, y 0, z 0

In Exercises 914, evaluate W f (x, y, z) dV for the function f and region W specified. 9.f (x, y, z) = z; W : x2 y 2, 0 x 1, x y z x + y

Calculate the integral of f (x, y, z) = z over the region W in Figure 11 below the hemisphere of radius 3 and lying over the triangle D

Calculate the integral of f (x, y, z) = ez over the tetrahedron W in Figure 12.

Integrate f (x, y, z) = x over the region in the first octant (x 0, y 0, z 0) above z = y2 and below z = 8 2x2 y2.

Compute the integral of f (x, y, z) = y2 over the region within the cylinder x2 + y2 = 4, where 0 z y.

Find the triple integral of the function z over the ramp in Figure

Find the volume of the solid in R3 bounded by y = x2, x = y2, z = x + y + 5, and z = 0.

Find the volume of the solid in the octant x 0, y 0, z 0 bounded by x + y + z = 1 and x + y + 2z = 1.

Calculate W y dV , where W is the region above z = x2 + y2 and below z = 5, and bounded by y = 0 and y = 1. 2

Evaluate W xz dV , where W is the domain bounded by the elliptic cylinder x2 4 + y2 9 = 1 and the sphere x2 + y2 + z2 = 16 in the first octant x 0, y 0, z 0 (Figure 14). FI

Describe the domain of integration and evaluate: 3 0 9x2 0 9x2y2 0 xy dz dy dx 25

Describe the domain of integration of the following integral: 2 2 4z2 4z2 5x2z2 1 f (x, y, z) dy dx dz 26

Let W be the region below the paraboloid x2 + y2 = z 2 that lies above the part of the plane x + y + z = 1 in the first octant (x 0, y 0, z 0). Express W f (x, y, z) dV as an iterated integral (for an arbitrary function f ). 2

In Example 5, we expressed a triple integral as an iterated integral in the three orders dz dy dx, dx dz dy, and dy dz dx Write this integral in the three other orders: dz dx dy, dx dy dz, and dy dx dz

Let W be the region bounded by y + z = 2, 2x = y, x = 0, and z = 0

Let W =  (x, y, z) :  x2 + y2 z 1  (see Figure 16). Express W f (x, y, z) dV as an iterated integral in the order dz dy dx (for an arbitrary function f ). F

Repeat Exercise 29 for the order dx dy dz.

Let W be the region bounded by z = 1 y2, y = x2, and the plane z = 0. Calculate the volume of W as a triple integral in the order dz dy dx.

Calculate the volume of the region W in Exercise 31 as a triple integral in the following orders: (a) dx dz dy (b) dy dz dx

In Exercises 3336, draw the region W and then set up but do NOT compute a single triple integral that yields the volume of W. The region W is bounded by the surfaces given by z = 1 y2, x = 0 and z = 0, z + x = 3

In Exercises 3336, draw the region W and then set up but do NOT compute a single triple integral that yields the volume of W. The region W is bounded by the surfaces given by z = x2, z + y = 1, and z y = 1.

In Exercises 3336, draw the region W and then set up but do NOT compute a single triple integral that yields the volume of W. The region W is bounded by the surfaces given by z = y2, y = z2 and x = 0, x + y + z = 4.

In Exercises 3336, draw the region W and then set up but do NOT compute a single triple integral that yields the volume of W. The region W is bounded by the surfaces given by z = 1 x2, z = 0, and y = 3 x2 z2.

In Exercises 3740, compute the average value of f (x, y, z) over the region W 37. f (x, y, z) = xy sin(z); W = [0, 1][0, 1][0, 1]

In Exercises 3740, compute the average value of f (x, y, z) over the region W 38. f (x, y, z) = xyz; W : 0 z y x 1

In Exercises 3740, compute the average value of f (x, y, z) over the region W 39. f (x, y, z) = ey ; W : 0 y 1 x2, 0 z x

In Exercises 3740, compute the average value of f (x, y, z) over the region W 40. f (x, y, z) = x2 + y2 + z2; W bounded by the planes 2y + z =1, x = 0, x = 1, z = 0, and y = 0

In Exercises 41 and 42, let I = 1 0 1 0 1 0 f (x, y, z) dV and let SN ,N ,N be the Riemann sum approximation SN ,N ,N = 1 N3 N i=1 N j=1 N k=1 f i N , j N , k N 41. Calculate SN ,N ,N for f (x, y, z) = ex2yz for N = 10, 20, 30. Then evaluate I and find an N such that SN ,N ,N approximates I to two decimal places. 42. C

Calculate SN ,N ,N for f (x, y, z) = sin(xyz) for N = 10, 20, 30. Then use a computer algebra system to calculate I numerically and estimate the error |I SN ,N ,N |.

Use Integration by Parts to verify Eq. (8).

Compute the volume An of the unit ball in Rn for n = 8, 9, 10. Show that Cn 1 for n 6 and use this to prove that of all unit balls, the five-dimensional ball has the largest volume. Can you explain why An tends to 0 as n ?