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Unformatted text preview: UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT235Y–CALCULUS II FALL-WINTER 2008-09 ASSIGNMENT 5. DUE ON FEBRUARY 12. PROBLEMS AND SOLUTIONS Problem 1. The rectangular prism pictured below has a rectangular base and a top that is a portion of the plane z = ax + by + c . The four vertical edges can have different lengths. Show by double integration that volume of prism = (area of base) × (average of the lengths of vertical edges) . This formula can be thought of as generalizing the formula for the area of a trapezoid. Solution: Let x and y be as in the following figure Then the volume of the rectangular prism is given by Volume = Z x Z y ( ax + by + c ) dydx = Z x axy + by 2 2 + cy y = y y =0 dx = Z x axy + by 2 2 + cy dx = ax 2 y 2 + bxy 2 2 + cxy y = y y =0 = ax 2 y 2 + bx y 2 2 + cx y . 1 On the other hand the area of base of the prism is x y and the average of the lengths of vertical edges is equal to 1 4 (( ax + c ) + ( ax + by + c ) + ( by + c ) + c ) = ax 2 + by 2 + c. Therefore, volume of prism = (area of base) × (average of the lengths of vertical edges) . Problem 2. (i) Prove that Z a sin x x dx = Z a 1 1 + x 2 dx + Z a sin x x- cos a + x sin a 1 + x 2 e- ax dx. Hint: Apply Fubini’s Theorem to the integral RR [0 ,a ] × [0 ,a ] e- xy sin xdA . (ii) Given that sin x x- cos a + x sin a 1 + x 2 ≤ 3 for all x and a with x 6 = 0, show that lim a →∞ Z a sin x x- cos a + x sin a 1 + x 2 e- ax dx = 0 . (iii) Use (i) and (ii) to prove that Z ∞ sin x x dx = π 2 Solution: (i) By Fubini’s Theorem ZZ [0 ,a ] × [0 ,a ] e- xy sin xdA = Z a Z a e- xy sin xdydx = Z a Z a e- xy sin xdxdy. (1) We have Z a Z...
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