quiz7_sol - Solution In polar coordinates the integrand...

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Date: Mar 16, 2011 MA3160-08 Quiz 7 No Calculators! Justify all answers! Name (print): Solutions 1. (5 pts.) A solid region has the shape of a pyramid with base in the plane z = 0 and sides formed by the three planes y = 0, y - x = 4 and 2 x + y + z = 4. Write an iterated integral (do not try to evaluate) which gives the mass of this pyramid if the density is δ ( x , y , z ). Solution 2.5 -4.5 -3 -2 -1 1 2 1 2 3 4 X Axis y-x=4 2x+y=4 (0, 4, 0) y=0 Y Axis The intersection of the plane 2 x + y + z = 4 with the plane z = 0 is the line 2 x + y = 4, and the intersection of the plane y - x = 4 with the plane z = 0 is the line y - x = 4. The two lines y - x = 4 and 2 x + y = 4 intersect in the point (0 , 4 , 0). The mass is given by M = Z W δ ( x , y , z ) dV , which leads to M = Z 4 y = 0 Z (4 - y ) / 2 x = y - 4 Z 4 - 2 x - y z = 0 δ ( x , y , z ) dzdxdy 2. (5 pts.) Convert the integral Z 1 x = 0 Z x y = - x x dydx to polar coordinates and then evaluate it.
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Unformatted text preview: Solution In polar coordinates the integrand changes to x = r cos and the area element becomes dydx = rdrd . The region of integration changes from y = x to = / 4, and the radius r ranges from r = 0 to the vertical line x = 1 which is r = 1 / cos . Therefore, the integral becomes Z / 4 =-/ 4 Z 1 / cos r = r cos rdrd = Z / 4 =-/ 4 " r 3 3 # 1 / cos cos d = Z / 4 =-/ 4 cos 3 cos 3 d = 1 3 [tan ] / 4-/ 4 Consequently, Z 1 x = Z x y =-x x dydx = Z / 4 =-/ 4 Z 1 / cos r = r 2 cos drd = 2 3...
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