M427Lfin08 - i and S be the part of the surface z = 9-x 2-y...

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M427L Summer 2008 Final Exam Name NO NOTES. NO CALCULATORS. 1. Rewrite Z 1 - 1 Z 1 x 2 Z 1 - y 0 f ( x,y,z ) dz dy dx with order of integration dx dy dz. 2. Evaluate Z 1 0 Z 1 - x 2 0 Z 1 - x 2 - y 2 0 ( x 2 + y 2 + z 2 ) 2 dz dy dx . 3. Tell whether F ( x,y,z ) = (2 xz +sin y ) ~ ı + x cos y~ + x 2 ~ k is conservative. If so, find a potential function for F . 4. Evaluate Z 1 0 Z 2 2 y 4cos( x 2 ) dx dy. 5. Use Lagrange multipliers to find the minimum value of f ( x,y,z ) = x 2 + y + z 2 subject to the constraint 2 x + y + 4 z = 6. 6. Find the curvature of c ( t ) = h t,t 2 , 1 - t 2 i at the point where t = 1. 7. Let C be the curve given by ~ r ( t ) = h t,t 2 ,t 3 i , 0 t 1, and let F ( x,y,z ) = xy~ ı + yz~ + zx ~ k . Evaluate Z C F d s . 8. Evaluate ZZ S yz dS , where S is the part of the plane x + y + z = 1 in the 1st Octant. 9. Evaluate I C xy dx + x 2 y 3 dy where C is the triangle with vertices (0 , 0), (1 , 0), and (1 , 2) oriented counter-clockwise. 10. Let E be the region bounded by z = 1 - x 2 , z = 0, y = 0, and y + z = 2. Let S be the surface of E . Express the flux of the vector field F ( x,y,z ) = xy~ ı + ( y 2 + e xz 2 ) ~ + sin( xy ) ~ k over S as a triple integral. 11. Let F ( x,y,z ) = h yz,xz,xy
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Unformatted text preview: i and S be the part of the surface z = 9-x 2-y 2 that lies above z = 5, oriented upward. Use Stokes Theorem to evaluate ZZ S ( F ) d S . 12. Find the area of the cap cut from the sphere x 2 + y 2 + z 2 = 2 by the cone z = p x 2 + y 2 . 13. Evaluate Z C x 2 dx + yz dy + ( y 2 / 2) dz along the line segment C joining (0 , , 0) and (0 , 3 , 4). 14. Find the area inside the loop of the curve x = t 2-3, y = ( t 3 / 3)-t . 15. Let C be the curve of the intersection of ( x-1) 2 + 4 y 2 = 16 and 2 x + y + z = 3, oriented counter-clockwise when viewed from above. Let F ( x,y,x ) = ( z 2 + y 2 + sin x 2 ) i + (2 xy + z ) j + ( xz + 2 yz ) k . Evaluate I C F d s . 16. Find a parametrization of the surface x 3 + 3 xy + z 2 = 2....
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