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mathematics
precalculus

Questions and Answers of Precalculus

Use Stokes’ Theorem to evaluate ʃʃS curl F • dS.F(x, y, z) = xyz i + xy j + x2yz k,S consists of the top and the four sides (but not the bottom) of the cube with vertices (±1, ±1, ±1),
Use Stokes’ Theorem to evaluate ʃʃS curl F • dS.F(x, y, z) = tan-1(x2yz2) i + x2y j + x2z2 k, S is the cone x = √y2 + z2 , 0 < x < 2, oriented in the direction of the positive x-axis
Use Stokes’ Theorem to evaluate ʃʃS curl F • dS.F(x, y, z) = zey i + x cos y j + xz sin y k, S is the hemisphere x2 + y2 + z2 = 16, y > 0, oriented in the direction of the positive y-axis
Use Stokes’ Theorem to evaluate ʃʃS curl F • dS.F(x, y, z) = x2 sin z i + y2 j + xy k, S is the part of the paraboloid z = 1 - x2 - y2 that lies above the xy-lane, oriented upward
A hemisphere H and a portion P of a paraboloid are shown. Suppose F is a vector field on R3 whose components have continuous partial derivatives. Explain why s = [[ curl F· dS || curl F· dS Н ZA
Let F be an inverse square field, that is, F(r) = cr/|r|3 for some constant c, where r = x i + y j + z k. Show that the flux of F across a sphere S with center the origin is independent of the radius
The temperature at a point in a ball with conductivity K is inversely proportional to the distance from the center of the ball. Find the rate of heat flow across a sphere S of radius a with center at
Use Gauss’s Law to find the charge enclosed by the cube with vertices (±1, ±1, ±1) if the electric field is E(x, y, z) = x i + y j + z k
Use Gauss’s Law to find the charge contained in the solid hemisphere x2 + y2 + z2 < a2, z > 0, if the electric field is E(x, y, z) = x i + y j + 2z k
Seawater has density 1025 kg/m3 and flows in a velocity field v = y i + x j, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward
A fluid has density 870 kgym3 and flows with velocity v = z i + y2 j + x2 k, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward
Let S be the part of the sphere x2 + y2 + z2 = 25 that lies above the plane z = 4. If S has constant density k, find (a) the center of mass and (b) the moment of inertia about the z-axis.
(a) Give an integral expression for the moment of inertia Iz about the z-axis of a thin sheet in the shape of a surface S if the density function is p.(b) Find the moment of inertia about the z-axis
Find the mass of a thin funnel in the shape of a cone z = √x2 + y2 , 1 < z < 4, if its density function is (x, y, z) = 10 - z.
Find the center of mass of the hemisphere x2 + y2 + z2 = a2, z > 0, if it has constant density.
Find a formula for ʃʃS F • dS similar to Formula 10 for the case where S is given by x = k( y, z) and n is the unit normal that points forward (that is, toward the viewer when the axes are drawn
Find a formula for ʃʃs F • dS similar to Formula 10 for the case where S is given by y = h(x, z) and n is the unit normal that points toward the left.
Find the flux ofF(x, y, z) = sin(xyz) i + x2y j + z2ex/5 k across the part of the cylinder 4y2 + z2 = 4 that lies above the xy-plane and between the planes x = -2 and x = 2 with upward
Find the value of ʃʃs x2y2z2 dS correct to four decimal places, where S is the part of the paraboloid z = 3 - 2x2 - y2 that lies above the xy-plane.
Find the exact value of ʃʃs xyz dS, where S is the surfacez = x2y2, 0 < x < 1, 0 < y < 2.
Evaluate ʃʃs (x2 + y2 + z2) dS correct to four decimal places, where S is the surface z = xey, 0 < x < 1, 0 < y < 1.
Evaluate the surface integral ʃʃS F • dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward)
Evaluate the surface integral ʃʃS F • dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward)
Evaluate the surface integral ʃʃS F • dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward)
Evaluate the surface integral ʃʃS F • dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward)
Evaluate the surface integral ʃʃS F • dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward)
Evaluate the surface integral ʃS F • dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward)
Evaluate the surface integral ʃʃS F • dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward)
Evaluate the surface integral ʃʃS F • dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward)
Evaluate the surface integral ʃʃS F • dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward)
Evaluate the surface integral ʃʃS F • dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward)
Evaluate the surface integral ʃʃS F • dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward)
Evaluate the surface integral ʃʃS F • dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward)
Evaluate the surface integral.ʃʃS (x2 + y2 + z2) dS,S is the part of the cylinder x2 + y2 = 9 between the planes z = 0 and z = 2, together with its top and bottom disks
Evaluate the surface integral.ʃʃS xz dS,S is the boundary of the region enclosed by the cylinder y2 + z2 = 9 and the planes x = 0 and x + y = 5
Evaluate the surface integral.ʃʃS (x + y + z) dS,S is the part of the half-cylinder x2 + z2 = 1, z > 0, that lies between the planes y = 0 and y = 2
Evaluate the surface integral.ʃʃS (x2z + y2z) dS, S is the hemisphere x2 + y2 + z2 = 4, z > 0
Evaluate the surface integral.ʃʃS y2 dS,S is the part of the sphere x2 + y2 + z2 = 1 that lies above the cone z = √x2 + y2
Evaluate the surface integral.ʃʃS x dS, S is the surface y = x2 + 4z, 0 < x < 1, 0 < z < 1
Evaluate the surface integral.ʃʃS y2z2 dS,S is the part of the cone y = √x2 + z2 given by 0 < y < 5
Evaluate the surface integral.ʃʃS z2 dS,S is the part of the paraboloid x = y2 + z2 given by 0 < x < 1
Evaluate the surface integral.ʃʃS y dS,S is the surface z = 2/3 (x3/2 + y3/2), 0 < x < 1, 0 < y < 1
Evaluate the surface integral.ʃʃS x dS,S is the triangular region with vertices (1, 0, 0), (0, -2, 0), and (0, 0, 4)
Evaluate the surface integral.ʃʃS xz dS, S is the part of the plane 2x + 2y + z = 4 that lies in the first octant
Evaluate the surface integral.ʃʃS x2yz dS,S is the part of the plane z = 1 + 2x + 3y that lies above the rectangle [0, 3] x [0, 2]
Evaluate the surface integral.ʃʃS (x2 + y2) dS, S is the surface with vector equation r(u, v) = (2uv, u2 - v2, u2 + v2), u2 + v2 < 1
Evaluate the surface integral.ʃʃS y dS, S is the helicoid with vector equation r(u, v) = (u cos v, u sin v, v), 0 < u < 1, 0 < v < π
Evaluate the surface integral.ʃʃS xyz dS, S is the cone with parametric equations x = u cos v, y = u sin v, z = u, 0 < u < 1, 0 < v < π/2
Evaluate the surface integral.ʃʃS (x + y + z) dS, S is the parallelogram with parametric equations x = u + v, y = u - v, z = 1 + 2u + v, 0 < u < 2, 0 < v < 1
Suppose that f (x, y, z) = g(√x2 + y2 + z2), where t is a function of one variable such that g(2) = -5. Evaluate ʃʃS f(x, y, z) dS, where S is the sphere x2 + y2 + z2 = 4.
Let H be the hemisphere x2 + y2 + z2 = 50, z > 0, and suppose f is a continuous function with f (3, 4, 5) = 7, f (3, -4, 5) = 8, f (-3, 4, 5) = 9, and f (-3, -4, 5) = 12.By dividing H into four
A surface S consists of the cylinder x2 + y2 = 1, -1 < z < 1, together with its top and bottom disks. Suppose you know that f is a continuous function with f (±1, 0, 0) = 2 f (0, ±1, 0) = 3 f
Let S be the surface of the box enclosed by the planes x = ±1, y = ±1, z = ±1. Approximate ʃʃS cos(x + 2y + 3z) dS by using a Riemann sum as in Definition 1, taking the patches Sij to be the
(a) Find a parametric representation for the torus obtained by rotating about the z-axis the circle in the xz-plane with center (b, 0, 0) and radius a < b. (b) Use the parametric equations
Find the area of the part of the sphere x2 + y2 + z2 = a2 that lies inside the cylinder x2 + y2 = ax. ZA х+ y
The figure shows the surface created when the cylinder y2 + z2 = 1 intersects the cylinder x2 + z2 = 1. Find the area of this surface.
Find the area of the part of the sphere x2 + y2 + z2 − 4z that lies inside the paraboloid z = x2 + y2.
(a) Show that the parametric equations x = a cosh u cos v, y = b cosh u sin v, z = c sinh u, represent a hyperboloid of one sheet.(b) Use the parametric equations in part (a) to graph the hyperboloid
(a) Show that the parametric equations x = a sin u cos v, y = b sin u sin v, z = c cos u, 0 < u < π, 0 < v < 2π, represent an ellipsoid.(b) Use the parametric equations in part (a) to
(a) Set up, but do not evaluate, a double integral for the area of the surface with parametric equations x = au cos v, y = bu sin v, z = u2, 0 < u < 2, 0 < v < 2.(b) Eliminate the
Find the exact area of the surface z = 1 + 2x + 3y + 4y2, 1 < x < 4, 0 < y < 1.
Find the area of the surface with vector equation r(u, v) = (cos3u cos3v, sin3u cos3v, sin3v), 0 < u < , 0 < v < 2. State your answer correct to four decimal places.
(a) Use the Midpoint Rule for double integrals (see Section 15.1) with six squares to estimate the area of the surface z = 1/(1 + x2 + y2), 0 < x < 6, 0 < y < 4.(b) Use a computer algebra
Find, to four decimal places, the area of the part of the surface z = (1 + x2)/(1 + y2) that lies above the square |x| + |y| < 1. Illustrate by graphing this part of the surface.
Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral.The part of the surface z = ln(x2 +
Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral.The part of the surface z = cos(x2
If the equation of a surface S is z = f (x, y), where x2 + y2 < R2, and you know that |fx| < 1 and | fy | < 1, what can you say about A(S)?
Find the area of the surface.The part of the sphere x2 + y2 + z2 = b2 that lies inside the cylinder x2 + y2 = a2, where 0 < a < b
Find the area of the surface.The surface with parametric equations x = u2, y = uv, z = 1/2v2, 0 < u < 1, 0 < v < 2
Find the area of the surface.The helicoid (or spiral ramp) with vector equation r(u, v) = u cos v i + u sin v j + v k, 0 < u < 1, 0 < v < π
Find the area of the surface.The part of the paraboloid y = x2 + z2 that lies within the cylinder x2 + z2 = 16
Find the area of the surface.The part of the surface x = z2 + y that lies between the planes y = 0, y = 2, z = 0, and z = 2
Find the area of the surface.The part of the surface z = xy that lies within the cylinder x2 + y2 = 1
Find the area of the surface.The part of the surface z = 4 - 2x2 + y that lies above the triangle with vertices (0, 0), (1, 0), and (1, 1)
Find the area of the surface.The surface z = 2/3 (x3/2 + y3/2), 0 < x < 1, 0 < y < 1
Find the area of the surface.The part of the cone z = √x2 + y2 that lies between the plane y = x and the cylinder y = x2
Find the area of the surface.The part of the plane x + 2y + 3z = 1 that lies inside the cylinder x2 + y2 = 3
Find the area of the surface.The part of the plane with vector equation r(u, v) = (u + v, 2 - 3u, 1 + u - v) that is given by 0 < u < 2, -1 < v < 1
Find the area of the surface.The part of the plane 3x + 2y + z = 6 that lies in the first octant
Find an equation of the tangent plane to the given parametric surface at the specified point. Graph the surface and the tangent plane.r(u, v) = (1 - u2 - v2) i - v j - u k; (-1, -1, -1)
Find an equation of the tangent plane to the given parametric surface at the specified point. Graph the surface and the tangent plane.r(u, v) = u2 i + 2u sin v j + u cos v k; u = 1, v = 0
Find an equation of the tangent plane to the given parametric surface at the specified point.r(u, v) = u cos v i + u sin v j + v k; u = 1, v = π/3
Find an equation of the tangent plane to the given parametric surface at the specified point.x = u2 + 1, y = v3 + 1, z = u + v; (5, 2, 3)
Find an equation of the tangent plane to the given parametric surface at the specified point.x = u + v, y = 3u2, z = u - v; (2, 3, 0)
(a) What happens to the spiral tube in Example 2 (see Figure 5) if we replace cos u by sin u and sin u by cos u?(b) What happens if we replace cos u by cos 2u and sin u by sin 2u?
A lamina with constant density p(x, y) = p occupies the given region. Find the moments of inertia Ix and Iy and the radii of gyration x̅̅ and Y̅̅.The part of the disk x2 + y2 < a2 in the first
A lamina with constant density p(x, y) = p occupies the given region. Find the moments of inertia Ix and Iy and the radii of gyration x̅̅ and Y̅̅.The triangle with vertices (0, 0), (b, 0), and
A lamina with constant density p(x, y) = p occupies the given region. Find the moments of inertia Ix and Iy and the radii of gyration x̅̅ and Y̅̅.The rectangle 0 < x < b, 0 < y < h
Find the mass and center of mass of the lamina that occupies the region D and has the given density function p.D is enclosed by the curves y = 0 and y = cos x, —п/2
Find the mass and center of mass of the lamina that occupies the region D and has the given density function p.D is bounded by the curves y = e2x, y = 0, x = 0, x = 1; (x, y) = xy
Find the mass and center of mass of the lamina that occupies the region D and has the given density function p.D is bounded by y = x + 2 and y = x2; (x, y) = kx2
Use a double integral to find the area of the region.The region enclosed by both of the cardioids r = 1 + cos θ and r = 1 - cosθ
Find the area of the region that lies inside both curves.r = sin 2θ , r = cos 2θ
Find the area of the region that lies inside both curves.r = 1 + cosθ, r = 1 - cosθ
Find the area of the region that lies inside both curves.r = 1 + cosθ, r = 1 2 cosθ
Find the area of the region that lies inside the first curve and outside the second curve.r = 3 sin θ, r = 2 - sinθ
Find the area of the region that lies inside the first curve and outside the second curve.r = 3 cosθ, r = 1 + cos θ
Find the area of the region that lies inside the first curve and outside the second curve.r = 1 + cos θ, r = 2 - cosθ
Find the area of the region that lies inside the first curve and outside the second curve.r2 = 8 cos 2θ, r = 2
Find the area of the region that lies inside the first curve and outside the second curve.r = 1 - sin θ, r = 1

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