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Math Help - Integration Proofs

  1. #1
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    Unhappy Integration Proofs

    I'm really lost on these ones

    (a) suppose f and g are continuous functions on [a,b] such that \int_a^b f = \int_a^b g. Prove that \exists x \in [a,b] such that f(x)=g(x)

    (b) suppose f is an integrable function on [a,b] and suppose that g is a function such that f(x)=g(x) except for finitely many x \in [a,b]. Show that g is integrable on [a,b] and \int_a^b f = \int_a^b g
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by akolman View Post
    I'm really lost on these ones

    (a) suppose f and g are continuous functions on [a,b] such that \int_a^b f = \int_a^b g. Prove that \exists x \in [a,b] such that f(x)=g(x)
    \int_a^b f = \int_a^b g

    \Rightarrow \int_a^b f - \int_a^b g = 0

    \Rightarrow \int_a^b f-g = 0

    now what can you say?

    (b) suppose f is an integrable function on [a,b] and suppose that g is a function such that f(x)=g(x) except for finitely many x \in [a,b]. Show that g is integrable on [a,b] and \int_a^b f = \int_a^b g
    Suppose g is the function zero. So that f(x) = 0 for all x \in [a,b] except for finitely many points y_1, \dots , y_N. show that \int_a^b f(x)~dx = 0.

    Once you do the above, apply the argument to the difference f - g, to get \int_a^b f - g = 0 \implies \int_a^b f = \int_a^b g (use part (a))
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