MAT235PS5-SOLNS - UNIVERSITY OF TORONTO DEPARTMENT OF...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 10/16/2009 for the course MATH MAT235 taught by Professor Recio during the Fall '08 term at University of Toronto- Toronto.

Page1 / 7

MAT235PS5-SOLNS - UNIVERSITY OF TORONTO DEPARTMENT OF...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online