20E_Final_A

20E_Final_A - 4-4 ± x-y ± 4(see the picture below Use the...

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MATH 20E FINAL Please answer the following questions. You will not get credit for answers un- less you demonstrate how you arrived at them. In short, please show all work. Numerical integrations by calculator will not be accepted. Problem #1 ( pts.) Consider a wire which is shaped like the 3D cork-screw curve c ( t ) = ( cos( t ) , sin( t ) , t ) for 0 t π . Suppose the mass density is given by f ( x, y, z ) = 2 + x 2 + xy (in units of mass per units of length). Please compute the mass of this wire. (Hint: This is just a path integral problem.) 1

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2 Problem #2 ( pts. total) Please compute the surface area of the portion of the graph of z = f ( x, y ) = 1 - x 2 - y 2 which lies over the unit disk x 2 + y 2 1.
3 Problem #3 ( pts.) Please compute the following line integrals: a) The line integral F · d of the vector-field F = ( y 2 , x 2 ) over the unit circle x 2 + y 2 = 1 in the plane. b) Compute the line integral F · d over the curve c ( t ) = ( t 2 , sin( t ) , t 2 π 2 ) between 0 t π , and where F = f with f ( x, y, z ) = sin( xze y ).

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4 Problem #4 ( pts. total) Consider the diamond in the plane

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Unformatted text preview: 4 ,-4 ± x-y ± 4 } (see the picture below). Use the change of variables x = u + v , y = v-u to compute the integral: I = ±± R (1 + x + y ) dxdy . (Hint: One of the main things you need to Fgure out is the limits of integration in the ( u, v ) coordinates. It helps to notice that the domain in the ( u, v ) coordinates will be a box-a ± u ± a and-a ± v ± a . You just need to Fnd the correct value of a .) 5 Problem #5 ( pts. total) Compute the outward fux ± S ± F · ˆ n dS oF the vector-±eld ± F = ( x 3 , y 3 , z ) over the entire surFace oF the cylinder S = { x 2 + y 2 ± 1 , z = 0 } ∪{ x 2 + y 2 = 1 , ± z ± 1 } ∪{ x 2 + y 2 ± 1 , z = 1 } . 6 Problem #6 ( pts. total) Please compute the outward fux oF the curl ± S ( ∇ × ± F ) · ˆ n dS over the upper spherical cap S = { x 2 + y 2 + z 2 = 1 , ± z } oF the vector-±eld ± F = ( xy, x 2 , xe z ) ....
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