Evaluate the triple integral using only geometric interpretation and symmetry. ∭C (4 + 5x2yz2) dV, where C is the cylindrical region x2 + y2 < 4, -2 < z < 2
> Let f be continuous on [0, 1] and let R be the triangular region with vertices (0, 0), (1, 0), and (0, 1). Show that f(x + y) dA = {, uf (u) du Jo
> Use spherical coordinates. Find the volume of the part of the ball ρ < a that lies between the cones φ = π/6 and φ = π/3.
> Evaluate the integral by making an appropriate change of variables. ∬R sin (9x2 + 4y2) dA, where R is the region in the first quadrant bounded by the ellipse 9x2 + 4y2 = 1
> Evaluate the integral by making an appropriate change of variables. ∬R cos (y-x)/(y+x) dA, where R is the trapezoidal region with vertices (1, 0), (2, 0), (0, 2), and (0, 1)
> Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7. Exercise 44 14.7 Exercise 44: Find the points on the surface y2 = 9 + xz that are closest to the origin.
> Use spherical coordinates. Evaluate ∭E y2 dV, where E is the solid hemisphere x2 + y2 + z2 < 9, y > 0.
> Use spherical coordinates. Evaluate ∭E (x2 + y2) dV, where E lies between the spheres x2 + y2 + z2 = 4 and x2 + y2 + z2 = 9.
> Use spherical coordinates. Evaluate ∬B (x2 + y2 + z2)2 dV, where B is the ball with center the origin and radius 5.
> Use the given transformation to evaluate the integral. ∬R y2 dA, where R is the region bounded by the curves xy = 1, xy = 2, xy2 = 1, xy2 = 2; u = xy, v = xy2. Illustrate by using a graphing calculator or computer to draw R.
> Use the given transformation to evaluate the integral. ∬R xy dA, where R is the region in the first quadrant bounded by the lines y = x and y = 3x and the hyperbolas xy = 1, xy = 3; x = u/v, y = v
> Use the given transformation to evaluate the integral. ∬R (x2 - xy + y2) dA, where R is the region bounded by the ellipse x2 - xy + y2 = 2; x = /2u – V2/3 v, y= /2u + /2/3 v
> Use the given transformation to evaluate the integral. ∬R x2 dA, where R is the region bounded by the ellipse 9x2 + 4y2 = 36; x = 2u, y = 3v
> Evaluate the triple integral. ∭T y2 dV, where T is the solid tetrahedron with vertices (0, 0, 0), (2, 0, 0), (0, 2, 0), and (0, 0, 2)
> Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7. Exercise 43. 14.7 Exercise 43: Find the points on the cone z2 = x2 + y2 that are closest to the point (4, 2, 0).
> Evaluate the triple integral. ∭E (x – y) dV, where E is enclosed by the surfaces z = x2 - 1, z = 1 - x2, y = 0, and y = 2
> Evaluate the triple integral. ∭E 6xy dV, where E lies under the plane z = 1 + x + y and above the region in the xy-plane bounded by the curves y = √(x ), y = 0, and x = 1
> Sketch the solid described by the given inequalities. 1 < ρ < 2, π/2 < φ < π
> Sketch the solid described by the given inequalities. ρ < 1, 0 < φ < π/6, 0 < θ < π
> Find the image of the set S under the given transformation. S is the disk given by u2 + v2 < 1; x = au, y = bv
> Write the equation in spherical coordinates. (a). x2 + y2 + z2 = 9 (b). x2 - y2 - z2 = 1
> Identify the surface whose equation is given. ρ = cos φ
> Identify the surface whose equation is given. ρ cos φ = 1
> Find the Jacobian of the transformation. x = u + vw, y = v + wu, z = w + uv
> Find the Jacobian of the transformation. x = uv, y = vw, z = wu
> Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7. Exercise 42 14.7 Exercise 42: Find the point on the plane x - 2y + 3z = 6 that is closest to the point (0, 1, 1).
> Find the Jacobian of the transformation. x = peq, y = qep
> Find the Jacobian of the transformation. x = s cos t, y = s sin t
> Find the Jacobian of the transformation. x = u2 + uv, y = uv2
> Find the Jacobian of the transformation. x = 2u + v, y = 4u - v
> The joint density function for a pair of random variables X and Y is (a). Find the value of the constant C. (b). Find P (X (c). Find P (X + Y Cx(1 + y) if 0 <x< 1, 0 < y < 2 f(x, y) = otherwise
> The figure shows the surface created when the cylinder y2 + z2 = 1 intersects the cylinder x2 + z2 = 1. Find the area of this surface. z. y
> Find the area of the finite part of the paraboloid y = x2 + z2 cut off by the plane y = 25. [Hint: Project the surface onto the xz-plane.]
> Evaluate the double integral by first identifying it as the volume of a solid. le (4 – 2y) dA, R= [0, 1] × [0, 1]
> Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order and explain why it’s easier. ∬D y2 exy dA, D is bounded by y = x, y = 4, x = 0
> (a). Find inequalities that describe a hollow ball with diameter 30 cm and thickness 0.5 cm. Explain how you have positioned the coordinate system that you have chosen. (b). Suppose the ball is cut in half. Write inequalities that describe one of the hal
> For what values of the number r is the function continuous on R3? (x + y + 2)' if (x, y, z) * (0, 0, 0) f(x, y, z) = {x? + y² + z? if (x, y, z) = (0, 0, 0)
> A solid-lies above the cone z = √(x^2 + y^2 ) and below the sphere x2 + y2 + z2 = z. Write a description of the solid in terms of inequalities involving spherical coordinates.
> Sketch the solid described by the given inequalities. ρ < 2, ρ < csc φ
> Sketch the solid described by the given inequalities. 2 < ρ < 4, 0 < φ < π/3, 0 < θ < π
> Sketch the solid described by the given inequalities. r2 < z < 8 - r2
> Write the equations in cylindrical coordinates. (a). x2 - x + y2 + z2 − 1 (b). z = x2 - y2
> A 20-ft-by-30-ft swimming pool is filled with water. The depth is measured at 5-ft intervals, starting at one corner of the pool, and the values are recorded in the table. Estimate the volume of water in the pool. 10 15 20 25 30 3 4 7 8 8 5 4 7 8 10
> Express D as a region of type I and also as a region of type II. Then evaluate the double integral in two ways. ∬x dA, D is enclosed by the lines y = x, y = 0, x = 1
> Graph the solid that lies between the surfaces z = e^(〖-x〗^2 ) cos (x2 + y2) and z = 2 - x2 - y2 for |x | < 1, |y | < 1. Use a computer algebra system to approximate the volume of this solid correct to four decimal places.
> If a, b, and c are constant vectors, r is the position vector xi + yj + zk, and E is given by the inequalities 0 (aBy)? 8|a· (b × c) | (а г) (b - г)(с г) dV —D E
> Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. f(x, y) = x² – y*; x² + y² = 1
> Use a computer algebra system to find the exact value of the integral ∬R x5y3exy dA, where R = [0, 1] × [0, 1]. Then use the CAS to draw the solid whose volume is given by the integral.
> Find the volume of the solid by subtracting two volumes. The solid enclosed by the parabolic cylinder y = x2 and the planes z = 3y, z = 2 + y
> Evaluate the triple integral using only geometric interpretation and symmetry. ∭B (z3 + sin y + 3) dV, where B is the unit ball x2 + y2 + z2 < 1
> Write five other iterated integrals that are equal to the given iterated integral. ∫_0^1 ∫_y^1 ∫_0^z f (x,y,z) dx dz dy
> When studying the spread of an epidemic, we assume that the probability that an infected individual will spread the disease to an uninfected individual is a function of the distance between them. Consider a circular city of radius 10 miles in which the p
> Evaluate the integral by changing to cylindrical coordinates. ∫_(-2)^2 ∫_(-√(4-y^2 ))^(√(4-y^2 ) ∫_(√(x^2+y^2 ) ^2 xz dz dx dy
> Evaluate the triple integral. ∭T xz dV, where T is the solid tetrahedron with vertices (0, 0, 0), (1, 0, 1), (0, 1, 1), and (0, 0, 1)
> Use a graphing device to draw the solid enclosed by the paraboloids z = x2 + y2 and z = 5 - x2 - y2.
> Use Lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter p is a square.
> A cylindrical shell is 20 cm long, with inner radius 6 cm and outer radius 7 cm. Write inequalities that describe the shell in an appropriate coordinate system. Explain how you have positioned the coordinate system with respect to the shell.
> Write the equations in cylindrical coordinates. (a). 2x2 + 2y2 - z2 = 4 (b). 2x - y + z = 1
> Describe in words the surface whose equation is given. r = 2
> Suppose that X and Y are independent random variables, where X is normally distributed with mean 45 and standard deviation 0.5 and Y is normally distributed with mean 20 and standard deviation 0.1. (a). Find P (40 < X < 50, 20 < Y < 25). (b). Find P (4(X
> Suppose X and Y are random variables with joint density Function f (x, y) = {_0^0.1 e-(0.5x+0.2y) if x > 0, y > 0 otherwise. (a). Verify that f is indeed a joint density function. (b). Find the following probabilities. (i). P Y > 1) (ii). P (X < 2, Y <
> Draw an example of a region that is (a). both type I and type II (b). neither type I nor type II
> Let V be the volume of the solid that lies under the graph of f (x, y) = √(52 - x^2 - y^2 ) and above the rectangle given by 2 < x < 4, 2 < y < 6. Use the lines x = 3 and y = 4 to divide R into sub rectangles. Let L and U be the Riemann sums computed us
> A long piece of galvanized sheet metal with width w is to be bent into a symmetric form with three straight sides to make a rain gutter. A cross-section is shown in the figure. (a). Determine the dimensions that allow the maximum possible flow; that is
> Express D as a region of type I and also as a region of type II. Then evaluate the double integral in two ways. ∬x y dA, D is enclosed by the curves y = x2, y = 3x
> Evaluate the given integral by changing to polar coordinates. ∬R arctan (y/x) dA, where R = {(x, y) | 1 < x2 + y2 < 4, 0 < y < x}
> Find the area of the surface. The part of the sphere x2 + y2 + z2 = 4z that lies inside the paraboloid z = x2 + y2
> Draw an example of a region that is (a). type I but not type II (b). type II but not type I
> Evaluate the double integral by first identifying it as the volume of a solid. le (2.x + 1) dA, R= {(x, y) | 0 < x < 2, 0 < y < 4}
> Evaluate the double integral by first identifying it as the volume of a solid. le /2 dA, R= {(x, y) | 2 < x < 6, –1 <y< 5}
> The contour map shows the temperature, in degrees Fahrenheit, at 4:00 pm on February 26, 2007, in Colorado. (The state measures 388 mi west to east and 276 mi south to north.) Use the Midpoint Rule with m = n = 4 to estimate the average temperature in Co
> Find the area of the surface. The part of the paraboloid z = 1 - x2 - y2 that lies above the plane z = 22
> Find the area of the surface. The part of the plane 3x + 2y + z = 6 that lies in the first octant
> Marine biologists have determined that when a shark detects the presence of blood in the water, it will swim in the direction in which the concentration of the blood increases most rapidly. Based on certain tests, the concentration of blood (in parts per
> Use a CAS to compute the iterated integrals ∫_0^1 ∫_0^1 (x-y)/(x+y)^3 dy dx and dx dy ∫_0^1 ∫_0^1(x-y)/(x+y)^3 . Do the answers contradict Fubini’s Theorem? Explain what is happening.
> Sketch the solid whose volume is given by the iterated integral. ∫_0^1 ∫_0^1〖(2 -x^2 - y^2)〗dx dy
> Graph the solid bounded by the plane x + y + z = 1 and the paraboloid z = 4 - x2 - y2 and find its exact volume. (Use your CAS to do the graphing, to find the equations of the boundary curves of the region of integration, and to evaluate the double integ
> Use geometry or symmetry, or both, to evaluate the double integral. ∬D (ax3 + by3 + √(a^2 - x^2 ) d dA, D = [-a, a] × [-b, b]
> Use geometry or symmetry, or both, to evaluate the double integral. ∬D (2 + x2y3 - y2 sin x) dA, D = {(x, y) | |x | + |y | < 1j
> Use geometry or symmetry, or both, to evaluate the double integral. (2x + 3y) dA, D is the rectangle 0 < x < a, 0 < y < b
> Use geometry or symmetry, or both, to evaluate the double integral. ∬D √(R^2-x^2-y^2 ) dA, D is the disk with center the origin and radius R
> Use geometry or symmetry, or both, to evaluate the double integral. ∬D (x + 2) dA, D = {(x, y) | 0 < y < √(9 - x^2 )
> Find the average value of the function f (x, y) = 1/√(x^2 + y^2 ) on the annular region a2 < x2 + y2 < b2, where 0 < a < b.
> Prove Property 11. Property 11: If m < f(x, y) < M for all (x, y) in D, then mA(D) f(x, y) dA < MA(D)
> (a). If your computer algebra system plots implicitly defined curves, use it to estimate the minimum and maximum values of f (x, y) = x3 + y3 + 3xy subject to the constraint (x – 3)2 + (y – 3)2 = 9 by graphical methods. (b). Solve the problem in part (a)
> An agricultural sprinkler distributes water in a circular pattern of radius 100 ft. It supplies water to a depth of e2r feet per hour at a distance of r feet from the sprinkler. (a). If 0 < R < 100, what is the total amount of water supplied per hour to
> Find the averge value of f over the region D. f (x, y) = xy, D is the triangle with vertices (0, 0), (1, 0), and (1, 3)
> Use Property 11 to estimate the value of the integral. ∬T sin4(x + y) dA, T is the triangle enclosed by the lines y = 0, y = 2x, and x = 1 Property 11: If m < f(x, y) < M for all (x, y) in D, then mA(D) f(x, y) dA < MA(D)
> Use Property 11 to estimate the value of the integral. ∬S √(4- x^2 y^2 ) dA, S = {(x, y) | x2 + y2 0} Property 11: If m < f(x, y) < M for all (x, y) in D, then mA(D) f(x, y) dA < MA(D)
> Find the approximate volume of the solid in the first octant that is bounded by the planes y = x, z = 0, and z = x and the cylinder y = cos x. (Use a graphing device to estimate the points of intersection.)
> Express D as a union of regions of type I or type II and evaluate the integral. ∬D x2 dA yA 1 (1, 1) D -1 1 -1
> (a). Find the region E for which the triple integral is a maximum. (b). Use a computer algebra system to calculate the exact maximum value of the triple integral in part (a). (1 – x? – 2y? – 3z²) dV – 32') dV
> Find the moments of inertia Ix, Iy, I0 for the lamina of Exercise 15. Exercise 15: Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides of length a if the density at any point is proportional to the square of
> Consider the problem of minimizing the function f (x, y) = x on the curve y2 + x4 - x3 = 0 (a piriform). (a). Try using Lagrange multipliers to solve the problem. (b). Show that the minimum value is f (0, 0) = 0 but the Lagrange condition ∆f (0, 0) = ∆t
> (a). In what way are the theorems of Fubini and Clairaut similar? (b). If f (x, y) is continuous on [a, b] × [c, d] and glx, v) = [" [ f(s, o) dt ds for a <x< b, c<y<d, show that gry = gyx = f(x, y).
> The joint density function for random variables X, Y, and Z is f (x, y, z) = Cxyz if 0 < x < 2, 0 < y < 2, 0 < z < 2, and f (x, y, z) = 0 otherwise. (a). Find the value of the constant C. (b). Find P (X < 1, Y < 1, Z < 1). (c). Find P (X + Y 1 Z < 1).
> Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order and explain why it’s easier. ∬D y dA, D is bounded by y = x - 2, x = y2
> (a). Use cylindrical coordinates to show that the volume of the solid bounded above by the sphere r2 + z2 = a2 and below by the cone z = r cot φ0 (or φ = φ0), where 0 (b). Deduce that the volume of the spherical wedge g
> (a). A cylindrical drill with radius r1 is used to bore a hole through the center of a sphere of radius r2. Find the volume of the ring-shaped solid that remains. (b). Express the volume in part (a) in terms of the height h of the ring. Notice that the v
> Graph the solid that lies between the surface z = 2xy/ (x2 + 1) and the plane z = x + 2y and is bounded by the planes x = 0, x = 2, y = 0, and y = 4. Then find its volume.
> Use a graphing device to draw a silo consisting of a cylinder with radius 3 and height 10 surmounted by a hemisphere.
