Weekly Assignment #3

Instructions:  integrations in problems 1 and 2 must be evaluated by hand. Show your work.

1. Average value

Let f(x)=x(cos(2x)+sin(3x)).

Find the average value of f(x) on [0,π].

Using a graph of f(x), approximate to at least two decimal places all the x-values between 0 and π for which f(x)=average value.

2. Area and Volume

Let R be the region in the plane between one arch of the curve y=cos(x)sin(x) and the x-axis.

Find a range of values of x for this region.

Find the area of R.

Find the volume of the solid of revolution obtained by rotating R about the x axis.

3. Application

When we die, our bodies become rigid (rigor mortis sets in). Niderkorn’s (1872) observations on 113 bodies provides the main reference database for the development of rigor mortis. One can fit a lovely model to this somewhat unlovely data, for the proportion p(t) of bodies in complete rigor mortis after t hours. It is illustrated in the graph below:

The model is (e^(-((26.28`)/(t^(2.39`))))): that is,

p[t_]:=E^(-26.28t^(-2.39))

a. Compute the average proportion of bodies in rigor mortis in the time interval from 3 to 5 hours after death (write the integral, but you may use your calculator or Mathematica to produce your answer!).

b. You are called to the scene of a battle, and find 100 bodies strewn over the landscape. At 9:37 p.m., 87 of the bodies show complete rigor mortis. Find the most likely time that the act occurred, as predicted by the model. (Show how you arrive at the answer: you should be able to find the solution analytically – that is, by hand, without recourse to the calculator).