Use spherical coordinates. Find the volume of the solid that lies within the sphere x2 + y2 + z2 = 4, above the xy-plane, and below the cone z = √(x^2+y^2 ).
> Find the volume of the solid that lies under the hyperbolic paraboloid z = 3y2 - x2 + 2 and above the rectangle R = [-1, 1] × [1, 2].
> Find the volume of the solid that lies under the plane 4x + 6y - 2z + 15 = 0 and above the rectangle R = {(x, y) | -1 < x < 2, -1 < y < 1j.
> Calculate the double integral. ∬R ye-xy dA, R = [0, 2] × [0, 3]
> Show that the maximum value of the function is a2 + b2 + c2. Hint: One method for attacking this problem is to use the Cauchy-Schwarz Inequality: |a ∙ b | (ах + by + c)? f(x, y) = х? + у? + 1
> Calculate the double integral. ∬R x/(1+xy) dA, R = [0, 1] × [0, 1]
> Calculate the double integral. ∬R x sin (x + y) dA, R = [0, π/6] × [0, ×/3]
> Calculate the double integral. ∬R tanθ/√(1-t^2 ) dA, R = { (θ, t) | 0 < θ < π/3, 0 < t < 1/2}
> (a). Use a Riemann sum with m = n = 2 to estimate the value of ∬R xe-xy dA, where R = [0, 2] × [0, 1]. Take the sample points to be upper right corners. (b). Use the Midpoint Rule to estimate the integral in part (a).
> Calculate the double integral. ∬R xy^2/(x^2+1) dA, R = {(x, y) | 0 < x < 1, -3 < y < 3}
> Calculate the double integral. ∬R (y + xy-2) dA, R = {(x, y) | 0 < x < 2, 1 < y < 2}
> Calculate the double integral. ∬R x sec2 y dA, R = {(x, y) | 0 < x < 2, 0 < y < π/4}
> Calculate the iterated integral ∫_0^1 ∫_0^1 √(s+t) ds ds
> Calculate the iterated integral ∫_0^1 ∫_0^1 v(u+v^2)^4 du dv
> Calculate the iterated integral ∫_0^1 ∫_0^1 xy √(x^2+y^2 ) dy dx
> Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7. Exercise 48 14.7 Exercise 48: Find the dimensions of the box with volume 1000 cm3 that has minimal surface area.
> In evaluating a double integral over a region D, a sum of iterated integrals was obtained as follows: Sketch the region D and express the double integral as an iterated integral with reversed order of integration. § Scx. y) da = {' [" f(x, y) dx dy
> Calculate the iterated integral ∫_0^3 ∫_0^(π/2) sin^3 φ dφ dt
> Find the averge value of f over the region D. f (x, y) = x sin y, D is enclosed by the curves y = 0, y = x2, and x = 1
> Calculate the iterated integral ∫_0^1 ∫_0^2 ye^(x-y) dx dy
> Calculate the iterated integral ∫_1^4 ∫_1^2 (x/y + y/x dy dx
> Calculate the iterated integral ∫_1^3 ∫_1^5 lny/xy dy dx
> Express D as a union of regions of type I or type II and evaluate the integral. ∬D y dA yA 1 x=y- y y= (x+ 1) -1 -1
> If R = [0, 4] × [-1, 2], use a Riemann sum with m = 2, n = 3 to estimate the value of ∬R (1 - xy2) dA. Take the sample points to be (a) the lower right corners and (b) the upper left corners of the rectangles.
> Evaluate the integral by reversing the order of integration. ∫_0^8 ∫_(∛y)^2e^(x^4 ) dx dy
> Evaluate the integral by reversing the order of integration. ∫_0^1 ∫_arcsiny^(π/2)cos x √(1 + 〖cos〗^2 x) dx dy
> Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7. Exercise 47 14.7 Exercise 47: Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r.
> The average value of a function f (x, y, z) over a solid region E is defined to be where V (E) is the volume of E. For instance, if is a density function, then ρ_(ave) is the average density of E. Find the average height of the points in th
> The average value of a function f (x, y, z) over a solid region E is defined to be where V (E) is the volume of E. For instance, if is a density function, then ρ_(¬ave) is the average density of E. Find the average value of the f
> Suppose X, Y, and Z are random variables with joint density function f (x, y, z) = Ce-(0.5x+0.2y+0.1z) if x > 0, y > 0, z > 0, and f (x, y, z) = 0 otherwise. (a). Find the value of the constant C. (b). Find P (X < 1, Y < 1). (c). Find P (X < 1, Y < 1, Z
> Evaluate the integral by reversing the order of integration. ∫_0^1 ∫_3y^3e^(x^2 ) dx dy
> If E is the solid of Exercise 18 with density function ρ (x, y, z) = x2 + y2, find the following quantities, correct to three decimal places. Exercise 18: Evaluate the triple integral. ∭E z dV, where E is bounded by the cylinder y2 + z2 = 9 and the pla
> Let E be the solid in the first octant bounded by the cylinder x2 + y2 = 1 and the planes y = z, x = 0, and z = 0 with the density function ρ (x, y, z) = 1 + x + y + z. Use a computer algebra system to find the exact values of the following quantities f
> Show that ∫_(-∞)^∞ ∫_(-∞)^∞ ∫_(-∞)^∞√(x^2+y^2+y^2 ) e^(-(x^2+y^2+z^2)) dx dy dz = 2π (The improper triple integral is defined as the limit of a triple integral over a solid sphere as the radius of the sphere increases indefinitely.)
> The surfaces ρ = 1 + 1/5 sin mθ sin nφ have been used as models for tumors. The “bumpy sphere” with m = 6 and n = 5 is shown. Use a computer algebra system to find the volume it enclo
> Assume that the solid has constant density k. Find the moment of inertia about the z-axis of the solid cone √(x^2 +y^2 ) < z < h.
> Assume that the solid has constant density k. Find the moment of inertia about the z-axis of the solid cylinder x2 + y2 < a2, 0 < z < h.
> (a). Show that ∫_0^1 ∫_0^1 ∫_0^1 1/(1-xyz) dx dy dz = ∑_(n-1)^∞ 1/n^3 (Nobody has ever been able to find the exact value of the sum of this series.) (b). Show that ∫_0^1 ∫_0^1 ∫_0^1 1/(1-xyz) dx dy dz = ∑_(n-1)^∞ (-1) ^(n-1)/n^3. Use this equation to e
> Evaluate the integral by changing to spherical coordinates ∫_(-2)^2 ∫_(-√(4-y^2))^(√(4-y^2)) ∫_(2-√(4-x^2-y^2))^(2+√(a^2-x^2-y^2) (x^2+y^2+z^2) ^(3/2) dz dy dx
> Evaluate the integral by changing to spherical coordinates ∫_(-a)^a ∫_(-√(a^2-y^2 ))^(√(a^2-y^2 ) ∫_(-√(a^2-x^2-y^2 ))^(√(a^2-x^2-y^2 ) (x^2 z+y^2 z+z^3) dz dx dy
> Use cylindrical or spherical coordinates, whichever seems more appropriate. (a). Find the volume enclosed by the torus ρ = sin φ. (b). Use a computer to draw the torus.
> Use cylindrical or spherical coordinates, whichever seems more appropriate. Evaluate ∭E z dV, where E lies above the paraboloid z = x2 + y2 and below the plane z = 2y. Use either the Table of Integrals (on Reference Pages 6–10) or a computer algebra syst
> Let D be the disk with center the origin and radius a. What is the average distance from points in D to the origin?
> Find the volume of the solid by subtracting two volumes. The solid under the plane z = 3, above the plane z = y, and between the parabolic cylinders y = x2 and y = 1 - x2
> Calculate the iterated integral ∫_0^(π/6) ∫_0^(π/2) (sin x+sin y ) dy dx
> Use cylindrical or spherical coordinates, whichever seems more appropriate. Find the volume and centroid of the solid E that lies above the cone z = √(x^2+y^2 ) and below the sphere x2 + y2 + z+ = 1.
> Use spherical coordinates. Find the mass and center of mass of a solid hemisphere of radius a if the density at any point is proportional to its distance from the base.
> The figure shows the region of integration for the integral ∫_0^1 ∫_(√x)^1 ∫_0^(1-y) f (x,y,z) dz dy dx Rewrite this integral as an equivalent iterated integral in the five other
> Use spherical coordinates. Let H be a solid hemisphere of radius a whose density at any point is proportional to its distance from the center of the base. (a). Find the mass of H. (b). Find the center of mass of H. (c). Find the moment of inertia of H ab
> Use spherical coordinates. (a). Find the centroid of the solid in Example 4. (Assume constant density K.) (b). Find the moment of inertia about the z-axis for this solid.
> Express the integral ∭E f (x, y, z) dV as an iterated integral in six different ways, where E is the solid bounded by the given surfaces. y = 4 - x2 - 4z2, y = 0
> 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
> 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
