Let E be the region in the first octant bounded betweenthe plane x + 2y + z = 4, the

In Exercises 14, find the triple integrals of the function over the region W. f(x, y, z) = x2 + 5y2 z, W is the rectangular box 0 x 2, 1 y 1, 2 z 3.

In Exercises 14, find the triple integrals of the function over the region W. h(x, y, z) = ax + by + cz, W is the rectangular box 0 x 1, 0 y 1, 0 z 2.

In Exercises 14, find the triple integrals of the function over the region W. f(x, y, z) = sin x cos(y+z), W is the cube 0 x , 0 y , 0 z .

In Exercises 14, find the triple integrals of the function over the region W. f(x, y, z) = exyz, W is the rectangular box with corners at (0, 0, 0), (a, 0, 0), (0, b, 0), and (0, 0, c).

Sketch the region of integration in Exercises 513. , 1 0 , 1 1 , 1x2 0 f(x, y, z) dz dx dy

Sketch the region of integration in Exercises 513. , 1 0 , 1 1 , 1z2 0 f(x, y, z) dy dz dx

Sketch the region of integration in Exercises 513. , 1 0 , 1 1 , 1x2 1x2 f(x, y, z) dz dx dy

Sketch the region of integration in Exercises 513. , 1 1 , 1 0 , 1z2 1z2 f(x, y, z) dy dz dx

Sketch the region of integration in Exercises 513. , 1 1 , 1x2 1x2 , 1x2z2 0 f(x, y, z) dy dz dx

Sketch the region of integration in Exercises 513. , 1 0 , 1z2 1z2 , 1x2z2 0 f(x, y, z) dy dx dz

Sketch the region of integration in Exercises 513. , 1 0 , 1y2 0 , 1x2y2 1x2y2 f(x, y, z) dz dx dy

Sketch the region of integration in Exercises 513. , 1 0 , 1z2 1z2 , 1y2z2 1y2z2 f(x, y, z) dx dy dz

Sketch the region of integration in Exercises 513. , 1 0 , 1z2 0 , 1x2z2 1x2z2 f(x, y, z) dy dx dz

In Problems 1418, decide whether the integrals are positive, negative, or zero. Let S be the solid sphere x2 + y2 + z2 1, and T be the top half of this sphere (with z 0), and B be the bottom half (with z 0), and R be the right half of the sphere (with x 0), and L be the left half (with x 0). , T e z dV

In Problems 1418, decide whether the integrals are positive, negative, or zero. Let S be the solid sphere x2 + y2 + z2 1, and T be the top half of this sphere (with z 0), and B be the bottom half (with z 0), and R be the right half of the sphere (with x 0), and L be the left half (with x 0). , B e z dV

In Problems 1418, decide whether the integrals are positive, negative, or zero. Let S be the solid sphere x2 + y2 + z2 1, and T be the top half of this sphere (with z 0), and B be the bottom half (with z 0), and R be the right half of the sphere (with x 0), and L be the left half (with x 0). , S sin z dV

In Problems 1418, decide whether the integrals are positive, negative, or zero. Let S be the solid sphere x2 + y2 + z2 1, and T be the top half of this sphere (with z 0), and B be the bottom half (with z 0), and R be the right half of the sphere (with x 0), and L be the left half (with x 0). , T sin z dV

In Problems 1418, decide whether the integrals are positive, negative, or zero. Let S be the solid sphere x2 + y2 + z2 1, and T be the top half of this sphere (with z 0), and B be the bottom half (with z 0), and R be the right half of the sphere (with x 0), and L be the left half (with x 0). , R sin z dV

Let W be the solid cone bounded by z = x2 + y2 and z = 2. For Problems 1927, decide (without calculating its value) whether the integral is positive, negative, or zero. - W y dV

Let W be the solid cone bounded by z = x2 + y2 and z = 2. For Problems 1927, decide (without calculating its value) whether the integral is positive, negative, or zero. - W x dV

Let W be the solid cone bounded by z = x2 + y2 and z = 2. For Problems 1927, decide (without calculating its value) whether the integral is positive, negative, or zero. - W z dV

Let W be the solid cone bounded by z = x2 + y2 and z = 2. For Problems 1927, decide (without calculating its value) whether the integral is positive, negative, or zero. - W xy dV

Let W be the solid cone bounded by z = x2 + y2 and z = 2. For Problems 1927, decide (without calculating its value) whether the integral is positive, negative, or zero. - W xyz dV

Let W be the solid cone bounded by z = x2 + y2 and z = 2. For Problems 1927, decide (without calculating its value) whether the integral is positive, negative, or zero. - W (z 2) dV

Let W be the solid cone bounded by z = x2 + y2 and z = 2. For Problems 1927, decide (without calculating its value) whether the integral is positive, negative, or zero. - W x2 + y2 dV

Let W be the solid cone bounded by z = x2 + y2 and z = 2. For Problems 1927, decide (without calculating its value) whether the integral is positive, negative, or zero. - W exyz dV

Let W be the solid cone bounded by z = x2 + y2 and z = 2. For Problems 1927, decide (without calculating its value) whether the integral is positive, negative, or zero. - W (z x2 + y2) dV

Find the volume of the region bounded by the planes z = 3y, z = y, y = 1, x = 1, and x = 2.

Find the volume of the region bounded by z = x2, 0 x 5, and the planes y = 0, y = 3, and z = 0.

Find the volume of the region in the first octant bounded by the coordinate planes and the surface x + y + z = 2.

A trough with triangular cross-section lies along the xaxis for 0 x 10. The slanted sides are given by z = y and z = y for 0 z 1 and the ends by x = 0 and x = 10, where x, y, z are in meters. The trough contains a sludge whose density at the point (x, y, z) is = e3x kg per m3. (a) Express the total mass of sludge in the trough in terms of triple integrals. (b) Find the mass.

Find the volume of the region bounded by z = x+y, z = 10, and the planes x = 0, y = 0.

In Problems 3338, write a triple integral, including limits of integration, that gives the specified volume. Between z = x + y and z = 1+2x + 2y and above 0 x 1, 0 y 2.

In Problems 3338, write a triple integral, including limits of integration, that gives the specified volume. Between the paraboloid z = x2 + y2 and the sphere x2 + y2 + z2 = 4 and above the disk x2 + y2 1.

In Problems 3338, write a triple integral, including limits of integration, that gives the specified volume. Between 2x + 2y + z = 6 and 3x + 4y + z = 6 and above x + y 1, x 0, y 0.

In Problems 3338, write a triple integral, including limits of integration, that gives the specified volume. Under the sphere x2 + y2 + z2 = 9 and above the region between y = x and y = 2x 2 in the xy-plane in the first quadrant.

In Problems 3338, write a triple integral, including limits of integration, that gives the specified volume. Between the top portion of the sphere x2 + y2 + z2 = 9 and the plane z = 2.

In Problems 3338, write a triple integral, including limits of integration, that gives the specified volume. Under the sphere x2 + y2 + z2 = 4 and above the region x2 + y2 4, 0 x 1, 0 y 2 in the xy-plane.

In Problems 3942, write limits of integration for the integral - W f(x, y, z) dV where W is the quarter or half sphere or cylinder shown. 39

In Problems 3942, write limits of integration for the integral - W f(x, y, z) dV where W is the quarter or half sphere or cylinder shown. 40

In Problems 3942, write limits of integration for the integral - W f(x, y, z) dV where W is the quarter or half sphere or cylinder shown. 41

In Problems 3942, write limits of integration for the integral - W f(x, y, z) dV where W is the quarter or half sphere or cylinder shown. 42

Find the volume of the region between the plane z = x and the surface z = x2, and the planes y = 0, and y = 3.

Find the volume of the region bounded by z = x + y, 0 x 5, 0 y 5, and the planes x = 0, y = 0, and z = 0.

Find the volume of the pyramid with base in the plane z = 6 and sides formed by the three planes y = 0 and y x = 4 and 2x + y + z = 4.

Find the volume between the planes z =1+ x + y and x + y + z = 1 and above the triangle x + y 1, x 0, y 0 in the xy-plane.

Find the volume between the plane x + y + z = 1 and the xy-plane, for x + y 2, x 0, y 0.

A solid shaped like a wedge of cheese has as its base the xy-plane, bounded by the x-axis, the line y = x and the line x + y = 1. Its sides are vertical, and its top is the plane x + y + z = 2. At any point, the density of the solid is four times the distance from the xy-plane. (a) Express the mass of the region in terms of triple integrals. (b) Find the mass.

Find the mass of a triangular-shaped solid bounded by the planes z = 1+ x, z = 1 x, z = 0, and with 0 y 3. The density is = 10 z gm/(cm)3, and x, y, z are in cm.

Find the mass of the solid bounded by the xy-plane, yzplane, xz-plane, and the plane (x/3) + (y/2) + (z/6) = 1, if the density of the solid is given by (x, y, z) = x+y.

Find the mass of the pyramid with base in the plane z = 6 and sides formed by the three planes y = 0 and y x = 4 and 2x + y + z = 4, if the density of the solid is given by (x, y, z) = y.

Let E be the solid pyramid bounded by the planes x + z = 6, x z = 0, y + z = 6, y z = 0, and above the plane z = 0 (see Figure 16.26). The density at any point in the pyramid is given by (x, y, z) = z grams per cm3, where x, y, and z are measured in cm. (a) Explain in practical terms what the triple integral - E z dV represents. (b) In evaluating the integral from part (a), how many separate triple integrals would be required if we chose to integrate in the z-direction first? (c) Evaluate the triple integral from part (a) by integrating in a well-chosen order.

(a) What is the equation of the plane passing through the points (1, 0, 0), (0, 1, 0), and (0, 0, 1)? (b) Find the volume of the region bounded by this plane and the planes x = 0, y = 0, and z = 0.

Problems 5456 refer to Figure 16.27, which shows triangular portions of the planes 2x + 4y + z = 4, 3x 2y = 0, z = 2, and the three coordinate planes x = 0, y = 0, and z = 0. For each solid region E, write down an iterated integral for the triple integral - E f(x, y, z) dV. E is the region bounded by y = 0, z = 0, 3x 2y = 0, and 2x + 4y + z = 4.

Problems 5456 refer to Figure 16.27, which shows triangular portions of the planes 2x + 4y + z = 4, 3x 2y = 0, z = 2, and the three coordinate planes x = 0, y = 0, and z = 0. For each solid region E, write down an iterated integral for the triple integral - E f(x, y, z) dV. E is the region bounded by x = 0, y = 0, z = 0, z = 2, and 2x + 4y + z = 4.

Problems 5456 refer to Figure 16.27, which shows triangular portions of the planes 2x + 4y + z = 4, 3x 2y = 0, z = 2, and the three coordinate planes x = 0, y = 0, and z = 0. For each solid region E, write down an iterated integral for the triple integral - E f(x, y, z) dV. E is the region bounded by x = 0, z = 0, 3x 2y = 0, and 2x + 4y + z = 4.

Figure 16.28 shows part of a spherical ball of radius 5 cm. Write an iterated triple integral which represents the volume of this region.

A solid region D is a half cylinder of radius 1 lying horizontally with its rectangular base in the xy-plane and its axis along the y-axis from y = 0 to y = 10. (The region is above the xy-plane.) (a) What is the equation of the curved surface of this half cylinder? (b) Write the limits of integration of the integral - D f(x, y, z) dV in Cartesian coordinates.

Set up, but do not evaluate, an iterated integral for the volume of the solid formed by the intersections of the cylinders x2 + z2 = 1 and y2 + z2 = 1.

Problems 6062 refer to Figure 16.29, which shows E, the region in the first octant bounded by the parabolic cylinder z = 6y2 and the elliptical cylinder x2 + 3y2 = 12. For the given order of integration, write an iterated integral equivalent to the triple integral - E f(x, y, z) dV dz dx dy

Problems 6062 refer to Figure 16.29, which shows E, the region in the first octant bounded by the parabolic cylinder z = 6y2 and the elliptical cylinder x2 + 3y2 = 12. For the given order of integration, write an iterated integral equivalent to the triple integral - E f(x, y, z) dV dx dz dy

Problems 6062 refer to Figure 16.29, which shows E, the region in the first octant bounded by the parabolic cylinder z = 6y2 and the elliptical cylinder x2 + 3y2 = 12. For the given order of integration, write an iterated integral equivalent to the triple integral - E f(x, y, z) dV dy dz dx

Find the average value of the sum of the squares of three numbers x, y, z, where each number is between 0 and 2.

Let E be the region in the first octant bounded between the plane x + 2y + z = 4, the parabolic cylinder x = 2y2, and the coordinate planes (see Figure 16.30). For each of the following orders of integration, write down an iterated integral equivalent to the triple integral - E f(x, y, z) dV. (a) dz dy dx (b) dy dz dx

Problems 6566 concern the center of mass, the point at which the mass of a solid body in motion can be considered to be concentrated. If the object has density (x, y, z) at the point (x, y, z) and occupies a region W, then the coordinates (x, y, z) of the center of mass are given by x = 1 m , W x dV y = 1 m , W y dV z = 1 m , W z dV where m = - W dV is the total mass of the body. A solid is bounded below by the square z = 0, 0 x 1, 0 y 1 and above by the surface z = x + y + 1. Find the total mass and the coordinates of the center of mass if the density is 1 gm/cm3 and x, y, z are measured in centimeters.

Problems 6566 concern the center of mass, the point at which the mass of a solid body in motion can be considered to be concentrated. If the object has density (x, y, z) at the point (x, y, z) and occupies a region W, then the coordinates (x, y, z) of the center of mass are given by x = 1 m , W x dV y = 1 m , W y dV z = 1 m , W z dV where m = - W dV is the total mass of the body. Find the center of mass of the tetrahedron that is bounded by the xy, yz, xz planes and the plane x + 2y + 3z = 1. Assume the density is 1 gm/cm3 and x, y, z are in centimeters.

Problems 6769 concern a rotating solid body and its moment of inertia about an axis; this moment relates angular acceleration to torque (an analogue of force). For a body of constant density and mass m occupying a region W of volume V , the moments of inertia about the coordinate axes are Ix = m V , W (y2 + z2 ) dV Iy = m V , W (x2 + z2 ) dV Iz = m V , W (x2 + y2 ) dV Find the moment of inertia about the z-axis of the rectangular solid of mass m given by 0 x 1, 0 y 2, 0 z 3.

Problems 6769 concern a rotating solid body and its moment of inertia about an axis; this moment relates angular acceleration to torque (an analogue of force). For a body of constant density and mass m occupying a region W of volume V , the moments of inertia about the coordinate axes are Ix = m V , W (y2 + z2 ) dV Iy = m V , W (x2 + z2 ) dV Iz = m V , W (x2 + y2 ) dV Find the moment of inertia about the x-axis of the rectangular solid a x a, b y b and c z c of mass m.

Problems 6769 concern a rotating solid body and its moment of inertia about an axis; this moment relates angular acceleration to torque (an analogue of force). For a body of constant density and mass m occupying a region W of volume V , the moments of inertia about the coordinate axes are Ix = m V , W (y2 + z2 ) dV Iy = m V , W (x2 + z2 ) dV Iz = m V , W (x2 + y2 ) dV Let a, b, and c denote the moments of inertia of a homogeneous solid object about the x, y and z-axes respectively. Explain why a + b> c.

In Problems 7071, explain what is wrong with the statement. Let S be the solid sphere x2 + y2 + z2 1 and let U be the upper half of S where z 0. Then - S f(x, y, z) dV = 2 - U f(x, y, z) dV .

In Problems 7071, explain what is wrong with the statement. . - 1 0 - x 0 - y 0 f(x, y, z) dz dy dx = - 1 0 - 1 y - x 0 f(x, y, z) dz dx dy

In Problems 7273, give an example of: A function f such that - R fdV = 7, where R is the cylinder x2 + y2 4, 0 z 3.

In Problems 7273, give an example of: A nonconstant function f(x, y, z) such that if B is the region enclosed by the sphere of radius 1 centered at the origin, the integral - B f(x, y, z) dx dy dz is zero.

Are the statements in Problems 7483 true or false? Give reasons for your answer. If (x, y, z) is mass density of a material in 3-space, then - W (x, y, z) dV gives the volume of the solid region W.

Are the statements in Problems 7483 true or false? Give reasons for your answer. The region of integration of the triple iterated integral - 1 0 - 1 0 - x 0 f dzdydx lies above a square in the xy-plane and below a plane.

Are the statements in Problems 7483 true or false? Give reasons for your answer. If W is the entire unit ball x2 + y2 + z2 1 then an iterated integral over W has limits - 1 0 - 1x2 0 - 1x2y2 0 f dz dy dx.

Are the statements in Problems 7483 true or false? Give reasons for your answer. The iterated integrals - 1 0 - 1x 0 - 1xy 0 f dzdydx and - 1 0 - 1z 0 - 1yz 0 f dxdydz are equal.

Are the statements in Problems 7483 true or false? Give reasons for your answer. The iterated integrals - 1 1 - 1 0 - 1x2 0 f dzdydx and - 1 0 - 1 0 - 1z 1z f dxdydz are equal.

Are the statements in Problems 7483 true or false? Give reasons for your answer. If W is a rectangular solid in 3-space, then - W f dV = - b a - d c - k e fdz dy dx, where a, b, c, d, e, and k are constants.

Are the statements in Problems 7483 true or false? Give reasons for your answer. If W is the unit cube 0 x 1, 0 y 1, 0 z 1 and - W f dV = 0, then f = 0 everywhere in the unit cube.

Are the statements in Problems 7483 true or false? Give reasons for your answer. If f>g at all points in the solid region W, then - W f dV > - W g dV.

Are the statements in Problems 7483 true or false? Give reasons for your answer. If W1 and W2 are solid regions with volume(W1) > volume(W2) then - W1 f dV > - W2 f dV

Are the statements in Problems 7483 true or false? Give reasons for your answer. Both double and triple integrals can be used to compute volume.