235-08y-t3-SOLNS - UNIVERSITY OF TORONTO DEPARTMENT OF...

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UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT 235 Y - CALCULUS II TEST #3. MARCH 19, 2009. PROBLEMS AND SOLUTIONS 1. a) (7 marks) Evaluate , where D is the triangular region with vertices (0 , 0), (0 , 1) and (1 , 1). 2 y D ed ∫∫ A Solution: Our region D can be expressed as 0 y 1 , 0 x y . Therefore, 22 2 2 1 11 00 0 0 y yy y y D e e dA e dx dy y e dy e e −− ⎡⎤ == = ⎢⎥ ⎣⎦ ∫ ∫ = . b) (8 marks) Evaluate 2 4 R x zdV ∫∫∫ , where R is the solid region bounded by the surface z = xy and the planes z = 0 , x = 1 and y = x . Solution: Our region R can be expressed as 0 x 1 , 0 y x , 0 z x y . Therefore, 1 2 2 4 0 00 0 44 2 2 xx y x x xy R 2 x zdzdydx x z dydx x y dydx = ∫ ∫ ∫ 43 7 0 33 x x y dx x dx 1 1 2 = .
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Page 2 2. a) (7 marks) Evaluate 22 2 () D x dA xy + ∫∫ , where D = { ( x , y ) | 1 x 2 + y 2 4 and 0 y x 3 } . Solution: Using polar coordinates, our region R can be expressed as 1 r 2 , 0 θ π / 3 . Therefore, 2/ 3 2 / 3 2 2 2 2 10 1 0 (c o s) 1 cos ( ) D xr r dA d dr dr d r r ππ θθ == + ∫ ∫ [] 2 /3 0 1 3 1 sin 4 r ⎡⎤ ⎢⎥ ⎣⎦ . b) (8 marks) Evaluate R x yd V + ∫∫∫ , where R is the part of the ball x 2 + y 2 + z 2 2 lying in the first octant. Solution: Using polar coordinates, our region R can be expressed as 0 ρ 2 , 0 / 2 , 0 φ / 2 . Notice also that x 2 + y 2 = 2 sin 2 , therefore, 2 4 3 00 0 s i n R x V ddd φθφρ += /2 2 5 3 2 0 0 0 0 4 cos (1 cos )sin cos 55 d ρφ θφ =− = + 2 3 1 5 = .
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235-08y-t3-SOLNS - UNIVERSITY OF TORONTO DEPARTMENT OF...

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