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Thread: 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 $\displaystyle \int_a^b f = \int_a^b g$. Prove that $\displaystyle \exists x \in [a,b]$ 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 $\displaystyle x \in [a,b]$. Show that g is integrable on [a,b] and $\displaystyle \int_a^b f = \int_a^b g$
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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 $\displaystyle \int_a^b f = \int_a^b g$. Prove that $\displaystyle \exists x \in [a,b]$ such that f(x)=g(x)
    $\displaystyle \int_a^b f = \int_a^b g$

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

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

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

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