16 MULTIPLE INTEGRALS
16 MULTIPLE INTEGRALS
16 MULTIPLE INTEGRALS
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GROUP WORK 3, SECTION <strong>16</strong>.5<br />
A Slick Model<br />
An oil tanker has leaked its entire cargo of oil into the middle of the Pacific Ocean, far from any island or<br />
continent. The oil has spread out in all directions in a thin layer on the surface of the ocean. The slick can<br />
be modeled by the two-dimensional density function K exp<br />
(−2 x2 + y 2 )<br />
w 2 ,wherew is a fixed constant and<br />
the origin of the xy-plane represents the location of the tanker. Assuming that none of the oil evaporates, the<br />
density function must account for all of the oil and hence can be interpreted as a probability distribution.<br />
1. Suppose w = 2. Find the value of K which ensures that K exp<br />
(−2 x2 + y 2 )<br />
w 2 is a probability distribution.<br />
2. Find the expected values μ x and μ y of x and y in the probability distribution from Problem 1. Interpret<br />
your answer geometrically.<br />
3. Find the radius of the circle centered at the origin which contains exactly 99% of the oil in the slick.<br />
903
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