Mathematics 136 -- Calculus 2

Discussion 2 -- Integrals and Net Change; Another Application

January 31, 2014

Background

Suppose we apply the Evaluation theorem to an integral of the form ∫ab f '(x) dx, where the integrand is the derivative of another function f(x). Then f(x) will play the role of the antiderivative, and we have

∫ab f '(x) dx = f(b) - f(a)

We will call this the net change of f(x) over the interval from a to b. For instance, as in last week's discussion, if f(t) represents the height of the balloon above the ground as a function of time, then f'(t) = v(t) is the balloon's vertical velocity, and
∫ab f '(t) dt = f(b) - f(a)

gives the net change in the height of the balloon from t = a to t = b.

Discussion Questions

  1. If w'(t) represents the rate of growth of a child in pounds per year, explain in words what the value of the integral
    ∫48 w '(t) dt

    represents. (No calculations -- just give a one sentence description of what the result would mean, in real world terms.)
  2. A honeybee population starts at 200 bees and increases at a rate of n'(t) bees per week from the start of the counting (t = 0 weeks) until t = 10 weeks. Write an expression for the number of bees in the population at the end of the 10 weeks using a definite integral (and anything else you need).
  3. If f(x) represents the slope of a trail at x miles from the start of the trail, what does the value of
    ∫26 f '(x) dx

    represent in real world terms?
  4. Read and work problem 69 in Section 5.3 of our text.
  5. Now look at the following graphs and data showing the distribution of family incomes in the US in 1968 and 2010.

    (Source: "http://2012books.lardbucket.org/books/...v2.../s22-01-income-inequality.html‎")

    It looks like everyone is doing better in 2010 than in 1968, right? But what are we ignoring if we just look at the income levels? In which year is the coefficient of inequality larger? Should we be concerned about this?

Assignment

Group write-ups due in class on Tuesday, February 4.