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 much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials.
> Electric charge is distributed over the disk x2 + y2 < 1 so that the charge density at (x, y) is (x, y) = √(x^2 + y^2 ) (measured in coulombs per square meter). Find the total charge on the disk.
> Use polar coordinates to find the volume of the given solid. Under the paraboloid z = x2 + y2 and above the disk x2 + y2 < 25
> Find the moments of inertia Ix, Iy, I0 for the lamina of Exercise 6. Exercise 6: Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ. D is the triangular region enclosed by the lines y = 0, y =
> Find the moments of inertia Ix, Iy, I0 for the lamina of Exercise 3. Exercise 3: Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ. D = {(x, y) | 1 < x < 3, 1 < y < 4}; ρ (x, y) = ky2
> A lamina occupies the region inside the circle x2 + y2 = 2y but outside the circle x2 + y2 = 1. Find the center of mass if the density at any point is inversely proportional to its distance from the origin.
> 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 the distance from the vertex opposite the hypotenuse.
> Find the center of mass of the lamina in Exercise 13 if the density at any point is inversely proportional to its distance from the origin. Exercise 13: The boundary of a lamina consists of the semicircles y = √(1 - x^2 ) and y = √(4 - x^2 ) together w
> The boundary of a lamina consists of the semicircles y = √(1 - x^2 ) and y = √(4 - x^2 ) together with the portions of the x-axis that join them. Find the center of mass of the lamina if the density at any point is proportional to its distance from the o
> The plane 4x - 3y + 8z = 5 intersects the cone z2 = x2 + y2 in an ellipse. (a). Graph the cone and the plane, and observe the elliptical intersection. (b). Use Lagrange multipliers to find the highest and lowest points on the ellipse.
> A lamina occupies the part of the disk x2 + y2 < 1 in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.
> Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ. D is enclosed by the curves y = 0 and y = cos x, -π/2 < x < π/2; ρ (x, y) = y
> Evaluate the given integral by changing to polar coordinates. ∬R sin (x2 + y2) dA, where R is the region in the first quadrant between the circles with center the origin and radii 1 and 3.
> Evaluate the given integral by changing to polar coordinates. ∬R (2x – y) dA, where R is the region in the first quadrant enclosed by the circle x2 + y2 = 4 and the lines x = 0 and y = x
> Evaluate the given integral by changing to polar coordinates. ∬D x2y dA, where D is the top half of the disk with center the origin and radius 5.
> Evaluate the iterated integral. ∫_0^1 ∫_0^(e^y) √(1+e^y) dw dv
> 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. 3 х -3+
> 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. yA -1 1 R -1
> The plane x + y + 2z = 2 intersects the paraboloid z = x2 + y2 in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin.
> Use polar coordinates to find the volume of the given solid. Bounded by the paraboloids z = 6 - x2 - y2 and z = 2x2 + 2y2
> Use polar coordinates to find the volume of the given solid. Above the cone z = √(x^2 + y^2 ) and below the sphere x2 + y2 + z2 = 1
> 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
> 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 = 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
