MAT235PS6-SOLNS - UNIVERSITY OF TORONTO DEPARTMENT OF...

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Unformatted text preview: UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT235YCALCULUS II FALL-WINTER 2008-09 ASSIGNMENT 6. DUE ON MARCH 12. PROBLEMS AND SOLUTIONS Problem 1. Find the limit: lim t 1 t ZZ ( x- t ) 2 +( y- t ) 2 1 p x 2 + y 2 dxdy. Solution. Making the change of variable x = t + tu , and y = t + tv in the double integral ZZ ( x- t ) 2 +( y- t ) 2 1 p x 2 + y 2 dxdy, we have ZZ ( x- t ) 2 +( y- t ) 2 1 p x 2 + y 2 dxdy = ZZ u 2 + v 2 1 t 2 p ( tu + t ) 2 + ( tv + t ) 2 ( x,y ) u,v dudv = t 3 ZZ u 2 + v 2 1 t 2 p ( u + 1) 2 + ( v + 1) 2 dudv. Hence, lim t 1 t ZZ ( x- t ) 2 +( y- t ) 2 1 p x 2 + y 2 dxdy = lim t t 2 ZZ u 2 + v 2 1 t 2 p ( u + 1) 2 + ( v + 1) 2 dudv. By the mean value theorem for double integrals, there is a point ( u, v ) in the disc u 2 + v 2 1 t 2 such that ZZ u 2 + v 2 1 t 2 p ( u + 1) 2 + ( v + 1) 2 dudv = p ( u + 1) 2 + ( v + 1) 2 ZZ u 2 + v 2 1 t 2 dudv = t 2 p ( u + 1) 2 + ( v + 1) 2 . (1) When t , ( u, v ) (0 , 0). Hence, lim t p ( u + 1) 2 + ( v + 1) 2 = 1 . Now making t approach infinity in equation (1) we conclude, lim t t 2 ZZ u 2 + v 2 1 t 2 p ( u + 1) 2 + ( v + 1) 2 dudv = lim t p ( u + 1) 2 + ( v + 1) 2 = . Therefore, lim t 1 t ZZ ( x- t ) 2 +( y- t ) 2 1 p x 2 + y 2 dxdy = . 1 Problem 2. Let us denote by V n the volume of the region in the first octant that lies beneath the surface x a 1 n + y b 1 n + z c 1 n = 1 , where a , b , c , and n are constants. (i) Find V 2 , (ii) Show that lim n V n = 0. Solution. Let us denote by B n the solid bounded by the surface ( x/a ) 1 n +( y/b ) 1 n +( z/c ) 1 n = 1, and the planes x = 0, y = 0, and z = 0. Then the volume of B n is given by the following triple integral V n = ZZZ B n dxdy dz. (2) Let x = au n , y = bv n and z = cw n . Since ( x/a ) 1 n + ( y/b ) 1 n + ( z/c ) 1 n = 1 we will have that u + v + w = 1. Consider the region of the space D = { ( u,v,w ) R 3 : u,v,w ,u + v + w 1 } , and the transformation T : B n D given by the equation T ( x,y,z ) = ( au n ,bv n ,cw n ) ....
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MAT235PS6-SOLNS - UNIVERSITY OF TORONTO DEPARTMENT OF...

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