Use symmetry to evaluate the double integral. ∬R (1 + x2 sin y + y2 sin x) dA, R = [- π, π ] × [-π , π]
> Use polar coordinates to find the volume of the given solid. Bounded by the paraboloid z = 1 + 2x2 + 2y2 and the plane z = 7 in the first octant
> Use polar coordinates to find the volume of the given solid. A sphere of radius a
> Use polar coordinates to find the volume of the given solid. Inside the sphere x2 + y2 + z2 = 16 and outside the cylinder x2 + y2 = 4
> Use polar coordinates to find the volume of the given solid. Below the plane 2x + y + z = 4 and above the disk x2 + y2 < 1
> 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. y. 1 R -1 1 x
> Evaluate the double integral. ∬D (x2 + 2y) dA, D is bounded by y = x, y = x3, x > 0
> Evaluate the double integral. ∬D x cos y dA, D is bounded by y = 0, y = x2, x = 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 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 = 1 + x2yey and the planes z = 0, x = 61, y = 0, and y = 1.
> 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?
