probset4 - : ] 1 , [ ] 1 , [ = = = otherwise prime q with q...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
_ _ University of Toronto DEPARTMENT OF MATHEMATICS MAT 237Y1Y Assignment #4 Due date: December 03, 6:10p.m Last Name: ____________________________ First Name: _______________________ Student #: _____________________________ Lecture Section: ______________ (a) TOTAL MARKS: 50 (b) WRITE SOLUTIONS ON THE SPACE PROVIDED, USE THE REVERSE SIDE OF A PAGE TO CONTINUE IF NECESSARY. (c) DO NOT REMOVE ANY PAGES. THERE ARE 6 PAGES (d) FOR A FULL MARK YOU MUST PRESENT YOUR SOLUTION CLEARLY MARKER'S REPORT Question Mark 1 /10 2 /12 3 /12 4 / 8 5 / 8 TOTAL /50 1
Background image of page 1

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

View Full DocumentRight Arrow Icon
1. Consider the set ,...} 3 , 2 , 1 : { \ ] 1 , 0 [ ], 1 , 0 [ : ) , {( 1 2 = = n y x R y x S n (a) Show that the outer area of S is 1 (i.e. ) ( S A = 1). (b) Prove that the inner area of S is 1 (i.e. 1 ) ( = S A ) and conclude that S is Jordan measurable . HINT: Show that for any 0 > ε there exist N subrectangles of R = that ] 1 , 0 [ ] 1 , 0 [ × cover S such that the sum of theirs areas is greater then 1 .
Background image of page 2
2. Consider the following function defined on the unit square R =
Background image of page 3

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

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

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

View Full DocumentRight Arrow Icon
Background image of page 6
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: : ] 1 , [ ] 1 , [ = = = otherwise prime q with q n m some for q n y and q m x if y x f , , 1 ) , ( (a) Is f integrable on R ? Justify your answer. (b) Do both iterated integrals exist? Justify your answer. 3. Verify that the assumptions of the Fubini Theorem are satisfied and evaluate (a) , where D dxdy y x ) , 2 max( ] 1 , [ ] 2 , [ = D . (b) , where D dxdy y E ) ( } 2 5 : ) , {( 2 2 = y x R y x D . Remark: denotes the integer part of the number u . ) ( u E 4. Evaluate dzdydx z z z x y 1 1 1 ) 2 ( ) sin( (Be careful: Why does the triple integral even exist?) 5. Let . Find vol ( } 1 ..... : { 1 2 1 = x x x x R n n n x ). Remark: Considering cases n = 2 and n = 3 first might be helpful....
View Full Document

Page1 / 6

probset4 - : ] 1 , [ ] 1 , [ = = = otherwise prime q with q...

This preview shows document pages 1 - 6. Sign up to view the full document.

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