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The xy plane that goes from the point 1 1 to the

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the xy-plane that goes from the point (-1,-1) to the point (1,1) along the curve y = x 5 against the force field defined by F ( x , y ) y x , yx 3 . W ______________________________________________________________________ 8. (10 pts.) Starting at the point (0,0), a particle goes along the y-axis until it reaches the point (0,4). It then goes from (0,4) to (-4,0) along the circle with equation x 2 + y 2 = 16. Finally the particle returns to the origin by travelling along the x-axis. Use Green’s Theorem to compute the work done on the particle by the force field defined by F ( x , y ) = < -5 y 3 , 5 x 3 > for ( x , y ) ε 2 . [Draw a picture. This is easy??]
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TEST4/MAC2313 Page 5 of 6 ______________________________________________________________________ 9. (10 pts.) (a) Show that the vector field F ( x , y ) < cos( x ) e y , sin( x ) e y 10 y > is actually a gradient field by producing a function φ ( x , y ) such that ∇φ ( x , y ) = F ( x , y ) for all ( x , y ) in the plane. (b) Using the Fundamental Theorem of Line Integrals, evaluate the path integral below, where C is any smooth path from the origin to the point ( π /2,ln(2)). [ WARNING: You must use the theorem to get any credit here.] C (cos( x ) e y ) dx (sin( x ) e y 10 y ) dy
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TEST3/MAC2313 Page 6 of 6 ______________________________________________________________________ 10. (10 pts.) Use the substitution u = (1/2)( x + y ) and v = (1/2)( x - y ) to evaluate the integral below, where R is the bound region enclosed by the triangular region with vertices at (0,0), (2,0), and (1,1). R cos 1 2 ( x y ) dA R cos 1 2 ( x y ) dA x , y ______________________________________________________________________ Silly 10 Point Bonus: Become a polar explorer. Reveal the details of how one can obtain the exact value of the definite integral 0 e x 2 dx even though there is no elementary anti-derivative for the function f( x ) e x 2 . Say where your work is, for it won’t fit here! [You may gloss some of the technical details related to the matter of convergence.]
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