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Additional Exercises 3

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Department
Mathematics
Course
MAT237Y1
Professor
All Professors
Semester
Fall

Description
1 1 1. [5 marks] Evaluate 1 dydx . 0 1 3+1 x Since the integral cannot be evaluated directly, we have to reverse the order of 1 3 integration. The region of integration is D = {( , ):x y 1, 0 x 1} , which can also be described as D = {(x y): 0 x y 3, 0 y 1}. Hence we have 1 1 1y3 1 3 1 dydx = 1 dxdy = y dy = 1 ln( y + 1)1 = 1 ln2 0 1 3 +1 y +1 y4 +1 4 0 4 x 00 0 (b) [6 marks] Suppose that the mass density d of the body occupying the region V in the first octant 2 bounded by the cylinder x = 4 y , the plane y + z =1, and the coordinate planes is given by d = f (x, y,z) where f is continuous on V. Write an iterated integral which gives the mass m of the body. Note that the plane y + z =1 intersects the xy-plane along the line y = 1. Projecting the region on xy-plane we have 1 4y 1 m = f (x,=y,z)dV f (x, y,z)dzdxdy V 0 0 0 1 1z 4y2 Projecting the region on the yz-plane we may write m = f (x, y,z)dxdydz . 0 0 0 Other answers possible.
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