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Trig. Substitution Problem

  1. Aug 11, 2011 #1
    1. The problem statement, all variables and given/known data
    the amount of nitrogen dioxide, a brown gas that impairs breathing, present in the atmosphere on a certain day in may in the city of long beach is appoximated by:

    A(t) = ((544) / (4 + (t - 4.5)^2) + 28 t is on interval [0, 11]

    where A(t) is measured in pollutant standard index (PSI) and t is measured in hours with t = 0 corresponding to 7 A.M. What is the average amount of the pollutant present in the atmosphere between 7 A.M. and 2 P.M. on that day in the city?

    2. Relevant equations
    I know that this problem can be solved using trig substitution since that is the section in my book, in which it came out from.

    3. The attempt at a solution
    I've made several attempts to solve this problem through integration/trig sub. but i keep getting the incorrect value.
  2. jcsd
  3. Aug 11, 2011 #2
    Shift the interval.

    ab f '(x-c) dx

    = f(x-c)|ab

    = f(b-c) -f(a-c)

    = ∫a-cb-c f '(x) dx

    After shifting, the trig-sub should be easy to detect.
    Last edited: Aug 11, 2011
  4. Aug 11, 2011 #3


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    I presume you know that the "average value" of an integrable function, f, between x= a and x= b, is
    [tex]\frac{\int_a^b f(x)dx}{b- a}[/tex]

    The "shifting" that Harrisonized refers to is equivalent to the simple substitution u= x- 4.5. Once you have done that your function will involve [itex]1/(4+ u^2[/itex] and you should be able to recognize that immediately.
  5. Aug 11, 2011 #4
    thank you so much, i knew i was missing something 'cause the integral itself doesnt find the average. so i was missing the b-a component. thanks i was able to figure it out.
  6. Aug 11, 2011 #5
    108 PSI if anyone was wondering.
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