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Questions and Answers of Calculus

Suppose the temperature of the solid in Figure begins at 40( at the bottom (the xy-plane) and increases (continuously) 5( for each unit above the xy-plane. Find the average temperature in the solid.
A full soda can of height h stands on the xy-plane. Punch a hole in the base and watch z̅ (the z-coordinate of the center of mass) as the soda leaks away. Starting at h/2, z̅ gradually drops to a
Let S = {(x, y, z): x2 / (2 + y2 / b2 + z2 / c2 ( 1}. Evaluate(xy + xz + yz) dV.
Suppose that the random variables (X, Y) have joint PDFFind each of the following: (a) k (b) P(Y > 4) (c) E(X)
Suppose that the random variables (X, Y, Z) have joint PDFFind each of the following: (a) k (b) P(X > 2) (c) E(X)
Suppose that the random variables (X, Y) have joint PDFFind each of the following: (a) P(X > 2) (b) P(X + Y ( 4) (c) E(X + Y)
Suppose that the random variables (X, Y) have joint probability density function ((x, y). The marginal probability density function of X is defined to beWhere ((x) and b(x) are the smallest and
In Problems 1-3, evaluate the integral which is given in cylindrical or spherical coordinates, and describe the region R of integration.1.2.3.
In Problems 1-3, use spherical coordinates to find the indicated quantity. 1. Mass of the solid inside the sphere ( = b and outside the sphere ( = ( (a < b) if the density is proportional to the
Find the volume of the solid bounded above by the plane z = y and below by the paraboloid z = x2 + y2. In cylindrical coordinates the plane has equation z = r sin ( and the paraboloid has equation z
Find the volume of the solid inside both of the spheres ( = 2(2 cos ( and ( = 2.
For any homogeneous solid S, show that the average value of the linear function ((x, y, z) = (x + by + cz + d on S is ((x̅, y̅, z̅), where (x̅, y̅, z̅) is the center of mass.
A homogeneous solid sphere of radius ( is centered at the origin. For the section S bounded by the half-planes ( = -( and ( = ( (like a section of an orange), find each value. (a) x-coordinate of the
All spheres in this problem have radius a, constant density k, and mass in. Find in terms of ( and m the moment of inertia of each of the following:(a) A solid sphere about a diameter(b) A solid
Suppose that the left sphere in Figure 10 has density k and the right sphere density ck. Find the y-coordinate of the center of mass of this two-sphere solid (convince yourself that the analogue of
In Problems 1-3, use cylindrical coordinates to find the indicated quantity.1. Volume of the solid bounded by the paraboloid z = x2 + y2 and the plane z = 4 2. Volume of the solid bounded above by
For the transformation x = u + v, y = v - u, sketch the u-curves and v-curves for the grid {(u, v): (u = 2, 3, 4, 5 and 1 ( v ( 3) or (v = 1, 2, 3 and 2 ( u ( 5)}.
In Problems 1-3, find the transformation from the uv-plane to the xy-plane and find the Jaeobian. Assume that x ( 0 and y ( 0.1. u = x + 2y, v = x - 2y2. u = 2x - 3y, v = 3x - 2y3. u = x2 + y2, v = x
For the transformation x = 2u + v, y = v - u, sketch the u-curves and v-curves for the grid {(u, v): (u = 2, 3, 4, 5 and 1 ( v ( 3) or (v = 1, 2, 3 and 2 ( u ( 5)}.
Find the Jacobian for the transformation from rectangular coordinates to spherical coordinates.
Find the volume of the ellipsoid x2/(2 + y2/b2 + z2/c2 = 1 by making the change of variables x = u(, y = vb, and z = cw. Also, find the moment of inertia of this solid about the z-axis assuming that
Suppose X and Y are continuous random variables with joint PDF ((x, y) and suppose U and V are random variables that are functions of X and Y such that the transformationX = x(U, V) and Y = y(U, V)is
Suppose that the random variables X and Y have joint PDFThat is, X and Y are uniformly distributed over the square 0 ( x ( 2, 0 ( y ( 2. Find (a) The joint PDF of U = X + Y and V = X - Y, and (b) The
Suppose X and Y have joint PDFFind (a) The joint PDF of U = X + Y and V = X (b) The marginal PDF of U.
For the transformation x = u sin v, y = it cos v, sketch the u-curves and v-curves for the grid {(u, v): (u = 0, 1, 2, 3 and 0 ( v ( () or (v = 0, (/2, ( and 0 ( u ( 3)}.
For the transformation x = u cos v, y = u sin v, sketch the u-curves and v-curves for the grid {(u, v): (u = 0, 1, 2, 3 and 0 ( v 2() or (v = 0, (, 2( and 0 ( u ( 3)}.
For the transformation x = u / (u2 + v2), y = -v/v2 + v2), sketch the u-curves and v-curves for the grid {(u, v): (u = 0, 1, 2, 3 and 1 ( v ( 3) or (v = 1, 2, 3 and -2 ( u ( 3)}.
For the transformation x = u + u / (u2 + v2), y = v - v / (u2 + v2), sketch the u-curves and v-curves for the grid {(u, v): (u = -2, -1, 0, 1, 2 and 1 ( v ( 3) or (v = 1, 2, 3 and -2 ( u ( 2)}.
In Problems 1-3, find the image of the rectangle with the given corners and find the Jacobian of the transformation.1. x = u + 2v, y = u - 2v; (0,0), (2,0),(2, 1), (0, 1)2. x = 2u + 3v, y = u - v;
In problem 1-4 sketch a sample of vectors for the given vector field F. 1. F(x, y) = xi + yj 2. F(x, y) = xi - yj 3. F(x, y) = - xi + 2yj 4. F(x, y) = 3xi + yj
In problem 1-3 find div F and curl F. 1. F(x, y, z) = x2i - 2xyj + yz2k 2. F(x, y, z) = x2i - y2j + z2k 3. F(x, y, z) = yzi - ezj + z2k
Let f be a scalar field and F a vector field. Indicate which of the following are scalar fields, vector fields, or meaningless. (a) Div f (b) Grad f (c) Curl F (d) Div (grad f) (e) Curl (grad f) (f)
Assuming that the required partial derivatives exist and are continuous, show that (a) Div (curl F) = 0; (b) Curl (grad f) = 0; (c) Div (fF) = (f) (div F) + (grad f) ∙ F; (d) Curl (fF) = (f)
Let F(x, y, z) = be an inverse square law field (Examples 3 and 4). Show that curl F = 0 and div F = 0.
Let F(x, y, z) = cr/||r||m, c ≠ 0, m ≠ 3. Show in contrast to Problem 21 that div F ≠ 0, though curl F = 0.
Let F(x, y, z) = f(r)r, where r ||r|| = √x2 + y2 ÷ z2) and f is a differentiable scalar function (except possibly at r = 0). Show that curl F = 0 (except at r = 0).
Let F(x, y, z) be as in Problem 23. Show that if div F = 0 then f(r) = cr-3, where c is a constant.
This problem relates to the interpretation of div and curl given in the margin box just after their definition. Consider the four velocity fields F, G, H, and L, which have for every z the
Sketch a plot of the vector field F = yi for (x, y) in the rectangle 1 ≤ x ≤ 2, 0 ≤ y ≤ 2. From the plot, use the marginal box that describes the interpretation of div and curl to determine
Sketch a plot of the vector field F = - x / (1 + x2 + y2)3/2 i - y / (1 + x2 + y2)3/2 j for (x, y) in the rectangle -1 s x s 1, -1 s y s 1. From the plot, use the marginal box that describes the
The scalar function div(grad 1) = ∇ ∙ ∇f (also written ∇2f) is called the Laplacian, and a function f satisfying ∇2f = 0 is said to be harmonic, concepts important in physics. Show that
Show that (a) Div (F × G) = G • curl F - F • curl G (b) Div (∇f × ∇g) = 0
By analogy with earlier definitions, define each of the following?(b) F(x, y, z) is continuous at (a, b, c)
In problem 1-5 find ▽f. 1. F(x, y, z) = x2 - 3xy + 2z 2. F(x, y, z) = sin (xyz) 3. F(x, y, z) = ln|xyz| 4. F(x, y, z) = ½ (x2 + y2 + z2) 5. F(x, y, z) = xey cos z 6. F(x, y, z) = y2e-2z
In problem 1-3 evaluate each line integral. 1. ∫C(x3 + y) ds; C is the curve x = 3t, y = t3, 0 ≤ t ≤ 1. 2. ∫Cxt2/5 ds; C is the curve x = ½ t, y = t5-2, 0 ≤ t ≤ 1. 3. ∫C(sin x + cos y)
A wire of constant density has the shape of the helix x = a cos t, y = a sin t, z = bt, 0 ≤ t ≤ 3π. Find its mass and center of mass.
In Problems 1-3, find the work done by the force field F in moving a particle along the curve C?1. F(x,y) = (x3 - y3)I + xy2j; C is the curve x = t2, y = t3, -1 ( t ( 0?2. F(x, y) = exi - e-yj; C is
Figure 9 shows a plot of a vector field F along with three curve, C1, C2, and C3. Determine whether each line integral (cF ( dx, i = 1, 2, 3, is positive, negative, or zero, and justify your answer?
Figure 10 shows a plot of a vector field F along with three curves, C1, C2, and C3. Determine whether each line integral (ci F ( dr, i = 1, 2, 3, is positive, negative, or zero, and justify your
Christy plans to paint both sides of a fence whose base is in the xy-plane with shape x = 30 cos3 t, y = 30 sin3 t, 0 ( t ( (/2, and whose height at (x, y) is 1 + 13 y, all measured in feet. Sketch a
A squirrel weighing 1.2 pounds climbed a cylindrical three by following the helical path x = cos t, y = sin t, z = 4t, 0 ( t ( 8( (distance measured in feet). How much did it do? Use a line integral,
Use a line integral to find the area of the part cut out of the vertical square cylinder |x| + |y| = a by the sphere x2 + y2 + z3 = z2. Check your answer by finding a trivial way to do this problem.
Use a line intergral to find the area of that part of the cylinder x2 + y2 = ay inside the sphere x2 = y2 + z2 = a2.
Two circular cylinders of radius a intersect so that their axes meet at right angles. Use a line integral to find the area of the part from one cut off by the other?
Evaluate(a) (cx2y ds using the parameterization x = 3 sin t, y = 3 cos t, 0 ( t ( (/2, which reverses and orientation of C I Example 1, and (b) (c4xy2 dx + xy2 by using the parameterization x = 3 -
In problem 1-4 determine whether the given field F is conservative. Is so, find f so that F = ∇f; if not, state that F is not conservative. 1. F(x, y) = (10x - 7y) i - (7x - 2y)j 2. F(x, y) =
In problem 1-2 show that the given line integral is independent of path (use Theorem C) and then evaluate the integral (either by choosing a convenient part or, if you prefer, by finding a potential
Foe each (x, y, z), let F(x, y, z) be a vector pointed toward the origin with magnitude inversely proportional to the distance from the origin; that is, letShow that F is conservative by finding a
For each (x, y, z), let F(x, y, z) be a vector pointed to ward the origin with magnitude inversely proportional to the distance from the origin; that is, letShow that F is conservative by finding a
Generalize Problems 22 and 23 by showing that if F(x, y, z) = [g(x2 + y2 + z2)] (xi + yj + zk) Where g is a continuous function of one variable, then F is conservative?
Suppose that an object of mass m is moved along a smooth curve C described byr = r (t) = x(t)i + y(t)j + z(t)k, a ‰¤ t ‰¤ bwhile subject only to the continuous force F. Show
Matt moved a heavy object along the ground from A to B. The object t was at rest at the beginning and at the end.
We normally consider the gravitational force of the earth on an object of mass m to be given by the constant F = - g mk, but, of course, this is valid only in regions near the earth's surface. Find
The distance from the earth (mass m) to the sun (mass M) varies from a maximum (aphelion) of 152.1 million kilometers to a minimum (perihelion) of 147.1 million kilometers. Assume that Newton's
This problem shows the need for simple connectedness in the "if" statement of Theorem C. Let F = (yi - xj) / (x2 + y2) on the set D = {(x, y): x2 + y2 ≠ 0}. Show each of the following. (a) The
Prove that in Theorem C, Condition (3) implies Condition (2)?
In Problems 1-3, use Green's Theorem to evaluate the given line integral. Begin by sketching the region S?1.Where C is the closed curve formed by y = x / 2 and y = (x between (0, 0) and (4, 2)?2.
Find the work done by F = (x2 + y2)i - 2xy j in moving a body counter clock wise around the curve C of Problem 14?
Use Green's Theorem to prove the plane case of Theorem 14.3D; that is, show that ˆ‚Nˆ‚/ˆ‚x = ˆ‚M/ˆ‚y implies thatWhich implies that F = Mi + Nj is conservative?
LetF = y = x2 + y2 i - x / x2 + y2 = j = Mi + Nj(a) Show that ˆ‚N/ˆ‚x = ˆ‚M/ˆ‚y.(b) Show, by using the parameterization x = cos t, y = sin t, thatWhere C is
Let F be as in Problem 19. CalculateWhere (a) C is the ellipse x2 / 9 + y2 / 4 = 1 (b) C is the square the vertices (1, - 1), (1, 1), (-1, 1), and (-1, -1). (c) C is the triangle with vertices (1,
Let the piecewise smooth, simple closed curve C be the boundary of a region S in the xy-plane. Modify the argument in Example 2 to show that
Calculate the work done by F = 2yi - 3xj in moving an object around the asteroid of Problem 23?
Let F(r) = r / ||r||2 = (xi + yj) / (x2 + y2).(a) Show that (c F ( n ds = 2(, where C is the circle centered at the origin of radius a and n = (xi + yj) / (x2 + y2 is the exterior unit normal to
Area of a Polygon Let V0(x0, y0), V1(x1, y1),....., Vn(xn, yn), be the vertices of a simple polygon P, labeled counter clock wise and with V0 = Vn. Show each of the following?(a) (c x dy = 1 / 2 (x1
In each of the following problems, plot the graph of f(x, y) and the corresponding gradient field F = (f on S = {(x, y)}; - 3 ( x ( 3, - 3 ( y ( 3}. In each case, curl F = 0 (Theorem 14.3D) and so
Let f(x, y) = 1n(cos (x / 3)) - In (cos (y / 3)). (a) Guess whether div F is positive or negative at a few points and then calculate div F to check on your guesses? (b) Calculate the flux of F across
Let f(x, y) = sin x sin y. (a) By visually examining the field F, guess where div F is positive and where it is negative. Then calculate div F to check on your guesses. (b) Calculate the flux of F
Let f(x,y) = exp(-(x2 + y2) / 4). Guess where div F is positive and where it is negative. Then determine this analytically?
In Problems 7 and 8, use the result of Example 2 to find the area of the indicated region S. Make a sketch. 1. S is bounded by the curves y = 4x and y = 2x2? 2. S is bounded by the curve y = 12 x3
In Problems 1-3, use the vector forms of Green's Theorem to calculate(a)and (b) 1. F = y2i + x2j; C is the boundary of unit square with vertices (0, 0), (1, 0), (1, 1), and (0, 1).2. F = ayi + bxj; C
In Problems 1-3, evaluate?1. g(x, y, z) = x2 + y2 + z; G: z = x + y + 1, 0 ( x ( 1, 0 ( y ( 12. g(x, y, z) = x; G: x + y + 2z = 4, 0 ( x ( 1, 0 ( y ( 13. g(x, y, z) = x + y; G: z = (4 - x2, 0 ( x (
Find the mass of the surface z = 1 - (x2 + y2) / 2 over 0 ( x ( 1, 0 ( y ( 1, if ((x, y, z) = kxy?
Find the center of mass of the homogeneous triangle with vertices (a, 0, 0), (0, a, 0), and (0, 0, a)?
In Problems 1-3, plot the parametric surface over the indicated domain? 1. r(u, v) = u i + 3v j + (4 - u2 - v2)k; 0 ( u ( 2, 0 ( v ( 1. 2. r(u, v) = 2u i + 3v j + (u2 + v2)k; -1 ( u ( 1, -2 ( v (
In Problems 1-4, se a CAS to plot of the parametric surface over the indicated domain and find the surface area of the resulting surface?1. r(u, v) = u sin v i + u cos u j + u k; - 6 ( u ( 6, 0 ( u (
Find the mass of the surface in Problem 23 if the density is proportional to the distance from the xy-plane?
Find the mass of the surface in Problem 24 if the density if proportional to (a) The distance from the z-axis, and(b) The distance from the xy-plane?
Show that the magnitude of the cross product ||ru (u, v) × rv (u, v)|| in Example 9 is equal to 25 |sin v|?
Let G be the sphere x2 + y2 + z2 = a2. Evaluate each of the following:(a)(b)(c) (d)(e)Use symmetry properties to make this a trivial problem?
The sphere x2 + y2 + z2 = a2 has constant area density k. Find each moment of inertia?(a) About a diameter(b) About a tangent line (Assume the Parallel Axis Theorem from Problem 28 of Section 13.5)
Find the total force against the surface of a tank full of a liquid of weight density k for each tank shape?a. Sphere of radius ab. Hemisphere of radius a with a flat basec. Vertical cylinder of
Find the center of mass of that part of the sphere x2 + y2 + z2 = a2 between the planes z = h1 and z = h2, where 0 ( h1 ( h2 ( a. Do this by the methods of this section and then compare with Problem
In Problems 1-3, use theorem B to calculate the flux of F across G?1. F(x, y, z) = - yi + xj; G is the part of the plane z = 8x - 4y - 5 above the triangle with vertices (0, 0, 0), (0, 1, 0), and (1,
In Problems 1-3, use Gauss's Divergence Theorem to calculate1. F(x, y, z) = z i + x j + y k; S is the hemisphere 0 ( z ( (9 - x2 - y2.2. F(x, y, z) = x i + 2y j + 3z k; S is the cube 0 ( x ( 1, 0 ( y
Use the result of Problem 15 to verify the formula for the volume of a right circular cylinder of height h and radius a?
Consider the plane ax + by + cz = d, where a, b, c, and d are all positive. Use Problem 15 to show that the volume of the tetrahedron cut from the first octant by this plane is d D / (3 (a2 + b2 +
CalculateIn each case, r = x i + y j + z k.(a) F = r/||r||3; S is the solid sphere (x - 2)2 + y2 + z2 ( 1,(b) F = r/||r||3; S is the solid sphere x2 + y2 + z2 ( a2,(c) F = r/||r||2; S as in part
We have defined the Laplacian of a scalar field byShow that if Dnf is the directional derivative in the direction of the unit normal vector n, then
Suppose that (2f is identically zero in a region S. Show that?

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