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review - Math 53 Spring 2000 sections 107 109 Review Sheet...

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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 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 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 , 0 x, 0 z 2 , z = 0 , 0 x, x 2 + y 2 1 , z = 2 , 0 x, x 2 + y 2 1 . 6. (Tuji Chang) Evaluate the line integral C y dx + z dy + x dz , where C consists of two line segments from (0 , 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 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.
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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 = x 2 + y 2 and below the sphere x 2 + y 2 + z 2 = 2. 14. (Albert Lee & Steve Hong) Evaluate 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. 15. (Albert Lee & Steve Hong) Find the (shortest) distance between the following two skew lines: 1 : x = 2 + t y = 5 - t z = 3 t 2 : x = 4 - s y = 6 s z = 7 s + 1 .
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