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Unformatted text preview: Oﬃce Problem 2  Math 139
Due midnight Dec. 10th Instructor
Oﬃce
Web page Mauro Maggioni
293 Physics Bldg.
www.math.duke.edu/˜ mauro/teaching.html In this Oﬃce Problem you will be constructing the Riemann integral for functions
f : Rn → R with n ≥ 1.
(i) You will start by doing Project 4 in Chapter 4 (page 161) of the textbook, which
takes you through the construction of the Riemann integral on 2dimensional rectangles.
(ii) After this, you can extend the deﬁnition of integral to higher dimensions: describe
how you would deﬁne the Riemann integral on ndimensional rectangles, i.e. sets
in the form R = [a1 , b1 ] × [a2 , b2 ] × · · · × [an , bn ], and discuss what parts of Project
4 would need to be modiﬁed, if any, in order to carry out this construction.
(iii) Finally, we may extend the deﬁnition of these higherdimensional Riemann integrals
to more general domains than rectangles. Explain in detail how you would deﬁne
the Riemann integral on a 2dimensional unit ball, i.e. on the set {(x, y ) ∈ R2 :
x2 + y 2 ≤ 1} ⊂ R2 : how would the reasonings in Project 4 change for such a
domain?
(iv) State a conjecture on which types of domains (besides rectangles and balls) to which
you think you could extend the deﬁnition of Riemann integrals. I will be meeting with you individually to answer your questions about this long
assignment: contact me by email with times that work for you and we will arrange to
meet. ...
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This note was uploaded on 01/16/2012 for the course MATH 139 taught by Professor Pardon,w during the Fall '08 term at Duke.
 Fall '08
 Pardon,W
 Math

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