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10.55. Find the area of the surface of the cone z2 3 x2 y2 cut out by the paraboloid z x2 y2 . Ans: 6 10.56. Find the surface area of the region common to the intersecting cylinders x2 y2 a2 and x2 z2 a2 . Ans: 16a2 p 10.57. (a) Obtain the surface area of the sphere x2 y2 z2 a2 contained within the cone z tan x2 y2 , 0 < < =2. (b) Use the result in (a) to nd the surface area of a hemisphere. (c) Explain why formally placing in the result of (a) yields the total surface area of a sphere. Ans: a 2 a2 1 cos ; b 2 a2 (consider the limit as ! =2 10.58. Determine the moment of inertia of the surface of a sphere of radius a about a point on the surface. Assume a constant density . Ans: 2Ma2 , where mass M 4 a2 10.59. (a) Find p the centroid of the surface of the sphere x2 y2 z2 a2 contained within the cone z tan x2 y2 , 0 < < =2. (b) From the result in (a) obtain the centroid of the surface of a hemisphere. Ans: a 1 a 1 cos ; b a=2 2 THE DIVERGENCE THEOREM 10.60. Verify the divergence theorem for A 2xy z i y2 j x 3y k taken over the region bounded by 2x 2y z 6; x 0; y 0; z 0. Ans: common value 27 10.61. Evaluate

To address the scaling problem, a hierarchical namespace methodology was adopted for the Internet community which enforces uniqueness and supports timely distribution of updates. This namespace system is a client/server protocol called BIND Domain Name Service (DNS). Computers and organizations in DNS are identified by a domain name. The domain name is a string tuple delimited by "."s which represents the administrative hierarchy of the namespace (Table 2.3). Domain names are usually represented in two to four levels. Note that there is no implied mapping of domain names to IP number octets or subnets. Table 2.3 Domain Names

%//N { 0.860521, 0.759653}

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8.19 A local telephone company wants to run a cable from point A on one side of a river 100 feet wide to point B on the opposite side, 500 feet along the shore from point C, which is opposite A. It costs three times as much money to run the cable underwater as on land. How should the company run the cable in order to minimize the cost of the project

A n dS, where A 2x 3z i xz y j y2 2z k and S is the surface of the sphere having Ans: 108

If we let a represent the cost per foot to run the cable on land, 3a is the cost to run a foot of cable underwater. The total cost is then c( x ) = 3a x 2 + 100 2 + a (500 x ) . Of course, 0 x 500.

100

Format: Examples:

c[x_]= 3 a x 2 + 1002 + a (500 x); Solve[c' [x] 0, x]

x dy dz y dz dx z dx dy, where S is the surface of the region bounded by the (a) by using the divergence theorem, (b) directly.

{{x 25 2}}

Now we must compare the cost corresponding to this solution with the cost at the endpoints of the interval. c[0] //N 800. a c[25 2] //N

The nicest functions to work with are polynomials. They are continuous and can easily be differentiated and integrated. If a difficult function is encountered in a problem, one approach is to approximate it by a polynomial.

4xz dy dz y2 dz dx yz dx dy, where S is the surface of the cube bounded by x 0, y 0, (a) directly, (b) By Green s theorem in space (divergence theorem).

A hierarchical name resolving protocol called BIND is used to recursively query domain name servers until a domain name is resolved to an IP number. In the DNS hierarchy (Figure 2.3), upper-level domain servers need only record the names of the next lower level of the name tuple along with the IP numbers of the name servers that resolve addresses for that level. All name servers must know the addresses of the top level Internet name servers which maintain a set of well-known top-level domain names (Table 2.4). The DNS system supports local management of the namespace and ensures timely information to the network at large. Table 2.4 Top Level Domains

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