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Unformatted text preview: Math 53, Spring 2000, sections 107 & 109 Review Sheet This file and others are available at http://www.math.berkeley.edu/hthall/53/ for download in .pdf format. Here are the problems which were submitted for extra credit (some of them have been slightly edited): 1. (Tommy Sy) Find the area of the portion of the plane 3 x +2 y +6 z = 6 which is bounded by the coordinate planes. 2. (Tommy Sy) Evaluate the integral ZZ R x- y x + y dA where R is the square with vertices (0 , 2), (1 , 1), (2 , 2), and (1 , 3). Hint : Use the transformation u = x- y , v = x + y . 3. (Eric Ng) The plane x + y + 2 z = 2 intersects the paraboloid z = x 2 + y 2 in an ellipse. Find the points on this ellipse that are closest to and furthest from the origin. 4. (Eric Ng) If z = f ( u, v ) where u = xy, v = y x , and f has continuous second derivatives, show that x 2 2 z x 2- y 2 2 z y 2 =- 4 uv 2 z u v + 2 v z v . 5. (Tuji Chang) Find RR S curl F d S , where F ( x, y,z ) = 2 xy i + xz j + yz k and S consists of the following three pieces, with outward orientation: x 2 + y 2 = 1 , x, z 2 , z = 0 , x, x 2 + y 2 1 , z = 2 , x, x 2 + y 2 1 . 6. (Tuji Chang) Evaluate the line integral R C y dx + z dy + xdz , where C consists of two line segments from (0 , , 0) to (1 , 1 , 2) and from (1 , 1 , 2) to (3 , 1 , 4). 7. (Allison Ryan) Prove that the maximum volume of a box with fixed surface area is achieved when all side lengths are equal. 8. (Allison Ryan) Find the mass of a sphere of radius a whose density (in the appropriate units) is equal to 2 r 2 , where r is the distance from the origin. 9. (Allison Ryan) Find the flux of the vector field F ( x, y,z ) = x,y 2 , 2 x + 5 through the cylinder x 2 + y 2 = 1 ,- 1 z 1, oriented outward. 10. (Bobby Young) Evaluate RR S curl F d S , where F ( x,y,z ) = xyz i + xy j + x 2 yz k and S is the top and four sides (but not the bottom) of the cube with vertices ( 1 , 1 , 1), oriented outward. 2 11. (Bobby Young) Find the area of the surface defined by the parametric equations x = uv,y = u + v,z = u- v over the region u 2 + v 2 1. 12. (Bobby Young) Show that every plane that is tangent to the cone x 2 + y 2 = z 2 passes through the origin. 13. (Albert Lee & Steve Hong) Find the volume of the solid that lies above the cone z = p x 2 + y 2 and below the sphere x 2 + y 2 + z 2 = 2. 14. (Albert Lee & Steve Hong) Evaluate R C F d r where F ( x,y, z ) = x i + y j + z k and C is the intersection of the surfaces z = x 2 + y 2 and x + 2 y + z = 6, oriented counter-clockwise as viewed from above....
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