Arnie Pizer Rochester Problem Library Fall 2005 WeBWorK assignment VMultIntegrals1Double due 04/03/2008 at 02:00am EST.
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1 WeBWorK assignment VMultIntegralsouble due 04/03/2008 at 02:00am ST.. ( pt) rochesterlibrary/setvmultintegralsouble/ur vc 8.pg Consider the solid that lies above the square = [0,2] [0,2] and below the elliptic paraboloid z = 36 x 2 2y 2. (A) stimate the volume by dividing into 4 equal squares and choosing the sample points to lie in the lower left hand corners. (B) stimate the volume by dividing into 4 equal squares and choosing the sample points to lie in the upper right hand corners.. (C) What is the average of the two answers from (A) and (B)? () Using iterated integrals, compute the exact value of the volume. 2. ( pt) rochesterlibrary/setvmultintegralsouble/ur vc 8 2.pg Z 4 Z 2 valuate the iterated integral 6x 2 y 3 dxdy ( pt) rochesterlibrary/setvmultintegralsouble/ur vc 8 3.pg Z 4 Z 2 valuate the iterated integral (4x + y) 2 dydx 3 4. ( pt) rochesterlibrary/setvmultintegralsouble/ur Z Z vc 8 4.pg Calculate the double integral (4x + 8y + 32) da where is the region: 0 x 4, 0 y ( pt) rochesterlibrary/setvmultintegralsouble/ur Z Z vc 8 5.pg Calculate the double integral xcos(2x + y) da where is the region: 0 x 2π 6, 0 y 2π 4 6. ( pt) rochesterlibrary/setvmultintegralsouble/ur vc 8 6.pg Calculate the volume under the elliptic paraboloid z = 4x 2 + 6y 2 and over the rectangle = [ 4, 4] [ 2, 2]. 7. ( pt) rochesterlibrary/setvmultintegralsouble/ur vc 8 7.pg Using geometry, calculate the volume of the solid under z = 49 x 2 y 2 and over the circular disk x 2 + y ( pt) rochesterlibrary/setvmultintegralsouble/ur vc 8 8.pg Using the maxima and minima of the function, produce upper and Zlower Z estimates of the integral I = e 5(x2 +y 2) da where is the circular disk: x 2 + y 2 4. I 9. ( pt) rochesterlibrary/setvmultintegralsouble/ur vc 8 9.pg Z Z +x valuate the iterated integral I = (9x 2 + 8y) dydx 0 x 0. ( pt) rochesterlibrary/setvmultintegralsouble/ur Z Z vc 8 0.pg valuate the double integral I = xyda where is the triangular region with vertices (0, 0),(, 0),(0, 4).. ( pt) rochesterlibrary/setvmultintegralsouble/ur vc 8.pg Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = ( pt) rochesterlibrary/setvmultintegralsouble/ur vc 8 2.pg valuate the integral by reversing the order of integration. Z Z 9 e x2 dxdy = 0 9y 3. ( pt) rochesterlibrary/setvmultintegralsouble/ur vc 8 3.pg Match the following integrals with the verbal descriptions of the solids whose volumes they give. Put the letter of the verbal description to the left of the corresponding integral Z 2 Z 4+ 4 x Z Z x 2 Z Z y 0 y Z 2 2 Z 2 4x + 3y dydx x 2 x2 y 2 dydx 4x 2 + 3y 2 dxdy 4 y 2 dydx 0 2 Z 3 Z 2 3y x 2 3y 2 dxdy A. One half of a cylindrical rod. B. Solid under a plane and over one half of a circular disk. C. Solid under an elliptic paraboloid and over a planar region bounded by two parabolas.. One eighth of an ellipsoid.. Solid bounded by a circular paraboloid and a plane. 4. ( pt) rochesterlibrary/setvmultintegralsouble/ur vc 8 4.pg Z 2 Z 5 If f (x) dx = 3 and g(x) dx = 3, what is the value of Z Z 3 2 f (x)g(y)da where is the square: 3 x 2, 2 y 5?
2 WeBWorK assignment VMultIntegrals2Polar due 04/04/2008 at 02:00am ST.. ( pt) rochesterlibrary/setvmultintegrals2polar/u VC 9.pg Z Z Using polar coordinates, evaluate the integral sin(x 2 + y 2 )da where is the region x 2 + y ( pt) rochesterlibrary/setvmultintegrals2polar/u VC 9 2.pg Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant between the circles x 2 + y 2 = 44 and x 2 2x + y 2 = ( pt) rochesterlibrary/setvmultintegrals2polar/u VC 9 3.pg Use the polar coordinates to find the volume of a sphere of radius ( pt) rochesterlibrary/setvmultintegrals2polar/u VC 9 4.pg A cylindrical drill with radius is used to bore a hole throught the center of a sphere of radius 3. Find the volume of the ring shaped solid that remains. 5. ( pt) rochesterlibrary/setvmultintegrals2polar/u VC 9 5.pg Z ZA. Using polar coordinates, evaluate the improper integral 2 e 5(x2 +y 2) dx dy. B. Use part A to evaluate the improper integral Z e 5x2 dx. 6. ( pt) rochesterlibrary/setvmultintegrals2polar/u VC 9 6.pg A sprinkler distributes water in a circular pattern, supplying water to a depth of e r feet per hour at a distance of r feet from the sprinkler. A. What is the total amount of water supplied per hour inside of a circle of radius 2? ft 3 /h B. What is the total amount of water that goes throught the sprinkler per hour? ft 3 /h
3 WeBWorK assignment VMultIntegrals3Appl due 04/05/2008 at 02:00am ST.. ( pt) rochesterlibrary/setvmultintegrals3appl/u VC 9 7.pg lectric charge is distributed over the disk x 2 + y 2 2 so that the charge density at (x,y) is σ(x,y) = 7 + x 2 + y 2 coulombs per square meter. Find the total charge on the disk. 2. ( pt) rochesterlibrary/setvmultintegrals3appl/u VC 9 8.pg A lamina occupies the part of the disk x 2 + y 2 9 in the first quadrant and the density at each point is given by the function ρ(x,y) = 2(x 2 + y 2 ). A. What is the total mass? B. What is the moment about the x-axis? C. What is the moment about the y-axis?. Where is the center of mass? (, ). What is the moment of inertia about the origin? 3. ( pt) rochesterlibrary/setvmultintegrals3appl/u VC 9 9.pg A lamp has two bulbs, each of a type with an average lifetime of 5 hours. The probability density function for the lifetime of a bulb is f (t) = 5 e t/5,t 0. What is the probability that both of the bulbs will fail within 2 hours? 4. ( pt) rochesterlibrary/setvmultintegrals3appl/u VC 9 0.pg You are getting married and your dearest relative has baked you a cake which fills the volume between the two planes, z = 0 and z = 2x + 0y + c, and inside the cylinder x 2 + y 2 =. You are to cut it in half by making two vertical slices from the center outward. Suppose one of the slices is at θ = 0 and the other is at θ = ψ. What is the limit, lim c ψ?
4 WeBWorK assignment VMultIntegrals4Surface due 04/06/2008 at 04:00am T.. ( pt) rochesterlibrary/setvmultintegrals4surface/ur vc 0.pg Find the surface area of the part of the plane 2x+5y+z = 5 that lies inside the cylinder x 2 + y 2 =. 2. ( pt) rochesterlibrary/setvmultintegrals4surface/ur vc 0 2.pg Find the surface area of the part of the circular paraboloid z = x 2 + y 2 that lies inside the cylinder x 2 + y 2 = ( pt) rochesterlibrary/setvmultintegrals4surface/ur vc 0 3.pg The vector equation r(u,v) = ucosvi+usinvj+vk, 0 v 4π, 0 u, describes a helicoid (spiral ramp). What is the surface area? 4. ( pt) rochesterlibrary/setvmultintegrals4surface/ur vc 0 4.pg Find the surface area of the surface of revolution generated by revolving the graph y = x 3, 0 x 7 around the x-axis.
5 WeBWorK assignment VMultIntegrals5Triple due 04/07/2008 at 02:00am T.. ( pt) rochesterlibrary/setvmultintegrals5triple/ur vc 0 5.pg valuate the triple integral xyzdv where is the solid: 0 z 3, 0 y z, 0 x y. 2. ( pt) rochesterlibrary/setvmultintegrals5triple/ur vc 0 6.pg Find the volume of the solid enclosed by the paraboloids z = 6 ( x 2 + y 2) and z = 2 6 ( x 2 + y 2). 3. ( pt) rochesterlibrary/setvmultintegrals5triple/ur vc 0 7.pg Find the average value of the function f (x,y,z) = x 2 + y 2 + z 2 over the rectangular prism 0 x, 0 y 4, 0 z 5 4. ( pt) rochesterlibrary/setvmultintegrals5triple/ur vc 0 8.pg Find the mass of the rectangular prism 0 x, 0 y 4, 0 z, with density function ρ(x,y,z) = x. You might find formula No. 3 on page 04 of the text helpful. 5. ( pt) rochesterlibrary/setvmultintegrals5triple/ur vc 0 9.pg Use cylindrical coordinates to evaluate the triple integral x 2 + y 2 dv, where is the solid bounded by the circular paraboloid z = 4 9 ( x 2 + y 2) and the xy -plane. 6. ( pt) rochesterlibrary/setvmultintegrals5triple/ur vc 0 0.pg Use spherical coordinates to evaluate the triple integral x 2 + y 2 + z 2 dv, where is the ball: x 2 + y 2 + z ( pt) rochesterlibrary/setvmultintegrals5triple/ur vc 0.pg Match the integrals with the type of coordinates which make them the easiest to do. Put the letter of the coordinate system to the left of the number of the integral.. dv where is: x 2 + y 2 + z 2 4, x 0, y 0, z 0 Z Z y 2 2. dx dy Z0Z Z0 x 3. z dv where is: x 2, 3 y 4, 5 zz Z 6 4. x 2 + y 2 da where is: x2 + y z 2 dv where is: 2 z 2, x 2 + y 2 2 A. cylindrical coordinates B. polar coordinates C. spherical coordinates. cartesian coordinates 8. ( pt) rochesterlibrary/setvmultintegrals5triple/ur vc 0 2.pg A volcano fills the volume between the graphs z = 0 and z = (x 2 +y 2 ) 29, and outside the cylinder x 2 + y 2 =. Find the volume of this volcano.
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