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265Lesson21Problems(2)

# 265Lesson21Problems(2) - (5 Evaluate the surface integral...

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MATH 265: MULTIVARIABLE CALCULUS: LESSON 21 PROBLEMS Problems are worth 20 points each. (1) Let S be the surface parametrized by Φ( u,v ) = ( u cos v,u sin v,u 2 + v 2 ). (a) Find a vector normal to S at the the point Φ(1 , 0) on the surface. (b) Find an equation for the plane tangent to S at the the point Φ(1 , 0) on the surface. (2) Find an equation for the plane tangent to the surface Φ( u,v ) = ( uv,u 2 v,v 2 ) at the point P = (2 , 2 , 4) on the surface. (3) Evaluate the surface integral ZZ D ( x + y )d S where D is the part of the plane x +2 y +3 z = 6 above the triangle in the xy -plane with vertices at (0 , 0 , 0) , (1 , 0 , 0), and (1 , 1 , 0). (4) Determine the area of the part of the paraboloid f ( x,y ) = a 2 - x 2 - y 2 ( a is a positive constant) above the xy -plane. Hint: Use polar coordinates.
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Unformatted text preview: (5) Evaluate the surface integral ZZ H y 2 z d S where H is the surface given parametri-cally by Φ( u,v ) = ( u cos v,u sin v,u ), 0 ≤ u ≤ 1, 0 ≤ v ≤ π . (Hint: Some of the grunge work for this problem is done in example 4, page 964, of the text.) (6) (Bonus question: worth 10 points. Total points for assignment not to exceed 100.) Consider the sphere x 2 + y 2 + z 2 = 4 a 2 of radius 2 a and center at the origin. Use a surface integral to ﬁnd a formula for the area of the cap , S , of the sphere above the disk x 2 + y 2 = a 2 in the xy-plane. (Hint: Use polar coordinates.) 1...
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