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Equating two integrals with a constant involved

  1. Dec 6, 2016 #1
    1. The problem statement, all variables and given/known data
    ##\int_{-1}^{3} f(x) dx = -4 = \int_{-1}^{3} 2g(x)dx##

    Now find a value(constant a) that makes the following true:

    ##\int_{-1}^{3} [3f(x) - ag(x) +a] dx = \int_{-1}^{3}(1-ax)dx##

    2. Relevant equations


    3. The attempt at a solution
    I'm unsure if my approach here is correct but I think that I need to utilize the fact that ##\int_{-1}^{3} f(x) dx## = -4 and substitute this in to the left side of equation? So I would get something like this after evaluating:

    ##\int_{-1}^{3} [-12+2a+a] dx## and then once i've done this, i can solve the integral. I'll have to solve the integral on the right side as well and once i've finished that, I find a value for a? I got 52/16 but I dont have an answer to check to.
     
  2. jcsd
  3. Dec 6, 2016 #2

    Mark44

    Staff: Mentor

    You can also use the fact that ##\int_{-1}^{3} 2g(x)dx = -4## to get an expression for ##\int_{-1}^{3} g(x)dx##.
    The two properties of definite integrals that come into play here are:
    ##\int_a^b k \cdot f(x) dx = k\int_a^b f(x) dx## (k a constant)
    and ##\int_a^b f(x) + g(x) dx = \int_a^b f(x) dx + \int_a^b g(x) dx##
     
  4. Dec 6, 2016 #3
    using these properties would I get something like this?(after splitting the left side):
    ##3\int_{-1}^{3} f(x)dx - a\int_{-1}^{3} g(x)dx + \int_{-1}^{3} adx## ?
     
  5. Dec 6, 2016 #4

    Mark44

    Staff: Mentor

    Yes, and you should be able to evaluate all three of these integrals, based on the given information.
     
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