Math 166
 Computer Lab # 6
  Applications of Integration 
  
  Name:_ _ _ _ _ _ _ _ _ _
Instructor:_ _ _ _ _ _ _ _ _

Goal:  To further investigate applications of integration

Problems

1.Greater Boston can be approximated by a semicircle of radius 8 miles with its center on the coast.  Moving away from the center along a radius, the population density is constant for the first mile.  Beyond that, the density starts to decrease according to the data given in the table, where g(r) is the population density in thousand people per square mile at a distance r miles from the center.

r (miles)     0     1     2     3     4     5     6     7     8

g(r)         75    75    67.5   60   52.5  45   37.5   30   22.5

Use this data and a Riemann sum to obtain 
  (i) an under-estimate for the total population of Boston living within the 8-mile limit, and 
  (ii) an over-estimate.  
  (iii) Use both of these estimates to obtain an improved estimate.

Hint:  If the population density was a constant for all 8 miles, then to get the total population we would just multiply the constant density times the area of the semicircle of radius 8 miles.  However, it is not constant over the entire 8 miles but from the data we are given we must assume that it is constant over each of the portions of the city where the radius varies from r=0 to r=1, from r=1 to r=2, from r=2 to r=3, etc.

2. In this problem we want to determine the volume of a typical hamburger and what percent of the volume is bread.  Suppose that a hamburger can be described as a solid of revolution and in particular can be formed by rotating the curve shown below about the y-axis.  The upper portion of the curve is given by y=f(x)=b-((b-c)/a9)x9 where a,b,c are the dimensions of the burger in inches.  In particular we choose a=2, b=1.7, and c=1.1 as indicated on the graph.

a=2;
b=1.7;
c=1.1; 
f[x_]:=b-(b-c)/(a^9)* x^9
p1=Plot[f[x],{x,0,a},PlotRange->{0,b},
  AspectRatio->Automatic,
  PlotStyle->RGBColor[1.000, 0.697, 0.139],
  Epilog->{{Hue[1.], Line[ {{0,f[2]}, {2,f[2]} } ],
   Line[{ {2,.35},{2,f[2]}}],Line[{{0,.35},{2,.35}}]}},
   DisplayFunction->Identity]
p2=ListPlot[{ {2,0},{2,.35}},PlotJoined->True,
  PlotStyle->RGBColor[1.000, 0.697, 0.139],
  DisplayFunction->Identity]
Show[p1,p2,Graphics[Text["burger",{1.0,.7}]],
  Graphics[Text["bun",{1.0,1.3}]],
  Graphics[Text["bun",{1.0,.2}]],DisplayFunction->$DisplayFunction,
  Ticks->{ {0,{2,"a"}},{0,{.35,"d"},{1.1,"c"},{1.7,"b"}}} 
 ]

(i) Find the total volume of the hamburger.

(ii) If the bun consists of the upper portion of the burger from y=c to y=f(x) and the lower portion from y=0 to y=d=.35 inches, find the percentage of the volume of the burger which is bread.

3. Suppose you wish to enclose a given area with some temporary flexible fence such as around a construction site. The region is sketched below.  The upper portion of the perimeter can be represented by the function y=12*ln(2x) from x=1 to x=91 where x represents feet. The remainder of the perimeter can be represented by the straight lines as shown.

Plot[12*Log[2x],{x,1, 91.},PlotRange->{0,70},
Epilog->{{Hue[1], Line[{{91,12Log[2]},{91,12Log[182]}}],
           Line[ { {1,12Log[2]},{91,12Log[2]}}] }},      
  PlotStyle->RGBColor[1.000, 0.137, 0.099],
  PlotLabel->"Field to be enclosed with fence" ]

Approximate the total length of fence needed to enclose the area.

4.  Two lanes of a running track are sketched below.  The equations for track 1 (inner red track) and track 2 (outer blue track) are given by

Clear[y1,y2]
y1[x_]:=Sqrt[80-.2*x^2]
y2[x_]:=Sqrt[88-.2*x^2]

p1=Plot[{y1[x] },{x,-19.8746,19.8746},DisplayFunction->Identity,
   PlotStyle->RGBColor[1.000, 0.086, 0.388]]
p2=Plot[{y2[x] },{x,-20.8567,20.8567},DisplayFunction->Identity,
   PlotStyle->RGBColor[0.131, 0.096, 1.000]]
p3=Plot[-x/4,{x,-19.,-18. },DisplayFunction->Identity]
p4=Plot[1,{x,-21,21},DisplayFunction->Identity]
Show[p1,p2,p3,p4,Graphics[Text["track 1",{19,2},{1,0}]], 
Graphics[Text["track 2",{-8.,9},{1,0}]],
Graphics[Text["starting point",{-21.,5.5},{-1,0}]],
Graphics[Text["y=1",{-8.,1.5},{1,0}]],
DisplayFunction->$DisplayFunction]

(i) Our first task is to find the length of each track.  We first need to obtain the x-values for our limits of integration.  Note that the inner track  begins and ends at the intersection of y1 and the line y=1;  similarly, the outer track begins and ends at the interseciton of y2 and the line y=1.   Find these points.

(ii) Try using the Integrate command to calculate the integral for the arc length for  the curve y1.  In this case Mathematica  gives the command back to you, indicating that it is unable to perform the calculation you asked it to do.

(iii) We  must turn to numerically approximating the lengths of the tracks by using the command NIntegrate which uses a numerical method such as Simpson's rule to approximate the integral.   Find the length of the inner and outer tracks using the command NIntegrate.

(iv) During a race, the starting point for track 1 is at the intercept of y1 and the line y=1 in the second quadrant.  The finishing points for both tracks are the positive intercepts of the curves and the line y=1.  Estimate where the starting point for the race (give x- and y- coordinates) should be placed on track 2 so that the two track lengths will be equal.  Give your answer so that the x-coordinate is to the nearest tenth of a foot.