Find the volume of the solid enclosed by the surface z = 1 + x2yey and the planes z = 0, x = 61, y = 0, and y = 1.
> Calculate the iterated integral ∫_0^1 ∫_0^1 (x+y)^2 dx dy
> Find the maximum and minimum volumes of a rectangular box whose surface area is 1500 cm2 and whose total edge length is 200 cm.
> Calculate the iterated integral ∫_1^4 ∫_0^2 (6x^2 y - 2x) dy dx
> Evaluate the given integral by changing to polar coordinates. ∬D x dA, where D is the region in the first quadrant that lies between the circles x2 + y2 = 4 and x2 + y2 = 2x
> Find ∫_0^2 f (x,y) dx and ∫_0^3 f(x,y) dy f (x, y) = x + 3x2y2
> Evaluate the given integral by changing to polar coordinates. ∬R cos √(x^2 + y^2 ) dA, where D is the disk with center the origin and radius 2
> Evaluate the given integral by changing to polar coordinates. ∬D e^(-x^2-y^2 ) dA, where D is the region bounded by the semicircle x = √(4 - y^2 ) and the y-axis.
> Evaluate the given integral by changing to polar coordinates. ∬R y^2/(x^2 + y^2 ) dA, where R is the region that lies between the circles x2 + y2 = a2 and x2 + y2 = b2 with 0 < a < b
> A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write ∬R f (x, y) dA as an iterated integral, where f is an arbitrary continuous function on R. T5 R 2 5 2.
> Evaluate the double integral. ∬D e^(-y^2 ) dA, D = {(x, y) | 0 < y < 3, 0 < x < y}
> Evaluate the double integral. ∬D (2x + y) dA, D = {(x, y) | 1 < y < 2, y - 1 < x < 1}
> Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7. Exercise 55 14.7 Exercise 55: If the length of the diagonal of a rectangular box must be L, what is the largest possible volume?
> Evaluate the iterated integral. ∫_0^1 ∫_0^(x^2)cos (s^3) dt ds
> Use a computer algebra system to find the exact volume of the solid. Under the surface z = x3y4 + xy2 and above the region bounded by the curves y = x3 - x and y = x2 + x for x > 0
> Sketch the solid whose volume is given by the iterated integral. ∫_0^1 ∫_0^(1-x^2) (1 - x) dy dx
> Evaluate the iterated integral. ∫_0^(π/2) ∫_0^x〖(x siny)〗dy dx
> Find the volume of the solid lying under the elliptic paraboloid x2/4 + y2/9 + z = 1 and above the rectangle R = [-1, 1] × [-2, 2].
> Find the volume of the solid by subtracting two volumes. The solid in the first octant under the plane z = x + y, above the surface z = xy, and enclosed by the surfaces x = 0, y = 0, and x2 + y2 = 4
> Calculate the double integral. ∬R 1/(1+ x+ y) dA, R = [1, 3] × [1, 2]
> Use a graphing calculator or computer to estimate the x-coordinates of the points of intersection of the curves y = x4 and y = 3x - x2. If D is the region bounded by these curves, estimate ∬D x dA.
> Find the volume of the given solid. Bounded by the cylinders x2 + y2 = r2 and y2 + z2 = r2
> Find the volume of the given solid. Bounded by the cylinder x2 + y2 = 1 and the planes y = z, x = 0, z = 0 in the first octant
> Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7. Exercise 52 14.7 Exercise 52: The base of an aquarium with given volume V is made of slate and the sides are made of glass. If slate costs five times as m
> Find the volume of the given solid. Bounded by the cylinder y2 + z2 = 4 and the planes x − 2y, x = 0, z = 0 in the first octant
> Evaluate the iterated integral. ∫_0^1 ∫_0^y (〖xe〗^(y^3 ) dx dy
> Find the volume of the given solid. Enclosed by the cylinders z = x2, y = x2 and the planes z = 0, y = 4
> Find the volume of the given solid. The tetrahedron enclosed by the coordinate planes and the plane 2x + y + z = 4
> Find the volume of the given solid. Enclosed by the paraboloid z = x2 + y2 + 1 and the planes x = 0, y = 0, z = 0, and x + y = 2
> Find the volume of the given solid. Under the surface z = xy and above the triangle with vertices (1, 1), (4, 1), and (1, 2)
> Find the volume of the given solid. Under the surface z = 1 + x2y2 and above the region enclosed by x = y2 and x = 4
> Find the volume of the given solid. Under the plane 3x + 2y - z = 0 and above the region enclosed by the parabolas y = x2 and x = y2
> Evaluate the double integral. ∬D y dA, D is the triangular region with vertices (0, 0), (1, 1), and (4, 0)
> Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7. Exercise 51 14.7 Exercise 51: Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is a constant c.
> Evaluate the double integral. ∬D y2 dA, D is enclosed by the quarter-circle y = √(1 -x^2 ), x > 0, and the axes
> Evaluate the iterated integral. ∫_0^2 ∫_0^2(x^2 y dx dy
> Evaluate the double integral. ∬D y2 dA, D is the triangular region with vertices (0, 1), (1, 2), (4, 1)
> Find ∫_0^2 f (x,y) dx and ∫_0^3 f(x,y) dy f (x, y) = y√(x + 2)
> The integral ∬R √(9 - y^2 ) dA, where R = [0, 4] × [0, 2], represents the volume of a solid. Sketch the solid.
> (a). Estimate the volume of the solid that lies below the surface z = xy and above the rectangle R = {(x, y) | 0 < x < 6, 0 < y < 4} Use a Riemann sum with m = 3, n = 2, and take the sample point to be the upper right corner of each square. (b). Use th
> Evaluate the double integral. ∬D y √(x^2- y^2 ) dA, D = {(x, y) | 0 < x < 2, 0 < y < x}
> Evaluate the iterated integral. ∫_1^5 ∫_0^x (8x - 2y) dy dx
> A contour map is shown for a function f on the square R = [0, 4] × [0, 4]. (a). Use the Midpoint Rule with m = n = 2 to estimate the value of ∬R f (x, y) dA. (b). Estimate the average value of f. yA 10 10 20 30 2. 10 2
> Use symmetry to evaluate the double integral. ∬R (1 + x2 sin y + y2 sin x) dA, R = [- π, π ] × [-π , π]
> Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7. Exercise 50 14.7 Exercise 50: Find the dimensions of the rectangular box with largest volume if the total surface area is given as 64 cm2.
> 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) = 3x + y; x? + y? = 10
> Use symmetry to evaluate the double integral. ∬R xy/(1+ x^4 ) dA, R = {(x, y) | -1 < x < 1, 0 < y < 1}
> Find the average value of f over the given rectangle. f (x, y) = ey √(x + e^y ), R = [0, 4] × [0, 1]
> Find the average value of f over the given rectangle. f (x, y) = x2y, R has vertices (-1, 0), (-1, 5), (1, 5), (1, 0)
> Find the volume of the solid in the first octant bounded by the cylinder z = 16 - x2 and the plane y = 5.
> Find the volume of the solid enclosed by the surface z = x2 + xy2 and the planes z = 0, x = 0, x = 5, and y = ±2.
> (a). Estimate the volume of the solid that lies below the surface z = 1 + x2 + 3y and above the rectangle R = [1, 2] × [0, 3]. Use a Riemann sum with m = n = 2 and choose the sample points to be lower left corners. (b). Use the Midpoint Rule to estimate
> 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.
