thq_09 - 16s MATH 2451 thq 09 Multivariable Calculus(Tue...

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VOID Exam : 16s MATH 2451 thq 09 Course : Multivariable Calculus Due Date : 2016-04-12 (Tue) Instructor : McCary T.C. Use (First Name) (Last Name) 2016-04-07 17:50

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VOID 1. (10 points) Be sure to clearly label every cube. (a) For each shaded cube C ~ k ,N determine its “address” (the values of ~ k and N ). (b) Shade the following cubes: C 2 - 3 , 1 and C 10 1 , 2 and C 3 4 , 0 1 - 1 1 - 1 2 - 2 2 - 2 3 - 3 3 - 3 4 - 4 4 - 4
VOID 2. (10 points) Recall the definition of the n -volume of a set: vol n ( A ) = Z R n A | d n x | A set A R 2 is shaded on the axis below. The 2-volume of A is defined as the integral above (with n = 2). (a) Compute L 1 ( A ) (b) Compute U 1 ( A ) 1 - 1 1 - 1 2 - 2 2 - 2 3 - 3 3 - 3 4 - 4 4 - 4

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VOID 3. (10 points) Recall that a set A R n is said to have n -volume zero if for every ε > 0, there is N > 0 such that the sum of the volume of all of the cubes C ∈ D N needed to cover A is less than ε . Let A = x x x (0 , 1) R 2 . In this exercise, you’ll show that A has 2-volume zero.
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