16.4 Prelim. Questions

# 16.4 Prelim. Questions - 936 C H A P T E R 16 M U LTI P L E...

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936 CHAPTER 16 MULTIPLE INTEGRATION (ET CHAPTER 15) A 5 = A 4 C 5 = π 2 2 · 16 15 = 8 2 15 5 . 26 The maximum value is A 5 . Combining with (3) we conclude that the fve-dimensional ball has the largest volume. By the closed Formulas For A 2 m and A 2 m + 1 , it Follows that these sequences tend to zero as n →∞ (Factorials grow Faster than exponentials). ThereFore, the sequence { A n } also tends to zero as n . 16.4 Integration in Polar, Cylindrical, and Spherical Coordinates (ET Section 15.4) Preliminary Questions 1. Which oF the Following represent the integral oF f ( x , y ) = x 2 + y 2 over the unit circle? (a) Z 1 0 Z 2 0 r 2 dr d θ (b) Z 2 0 Z 1 0 r 2 (c) Z 1 0 Z 2 0 r 3 (d) Z 2 0 Z 1 0 r 3 SOLUTION The unit circle is described in polar coordinates by the inequalities 0 2 , 0 r 1 Using double integral in polar coordinates, we have ZZ D f ( x , y ) dA = Z 2 0 Z 1 0 ³ ( r cos ) 2 + ( r sin ) 2 ´ rdrd = Z 2 0 Z 1 0 r 2 ³ cos 2 + sin 2 ´ = Z 2 0 Z 1 0 r 3 ThereFore (d) is the correct answer.

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16.4 Prelim. Questions - 936 C H A P T E R 16 M U LTI P L E...

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