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Math Help - Riemann integral

  1. #1
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    Riemann integral

    f(x)=\begin{cases}& \sin(x) , x\leq \pi /2  \\ & \1 , x> \pi /2\end{cases}

    Determine the riemann integral for every x\in R

    I(x)=\int_{0}^{x}f(t)dt



    I dont know if i do this right, but this is how i tried:

    I'(x)=f(x) according to fundamental theorem of calculus

    then I(x)=F(x) + C

    * if x\leq \pi /2


    I(x)=\int_{0}^{x}sin(x)dx = 1-cos(x)

    * if  x> \pi /2

    \int_{0}^{\pi/2}sin(x)dx + \int_{\pi/2+\delta }^{x}1dx=1+x-\frac{\pi}{2},\delta  \to \0


    so did i do totally wrong or does this look good? :P

    Thanks!
    Last edited by mechaniac; November 28th 2011 at 07:31 AM.
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  2. #2
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    Re: Riemann integral

    Quote Originally Posted by mechaniac View Post
    f(x)=\begin{cases}& \sin(x) , x\leq \pi /2  \\ & \1 , x> \pi /2\end{cases}
    I(x)=\int_{0}^{x}f(t)dt

    I'(x)=f(x) according to fundamental theorem of calculus
    then I(x)=F(x) + C
    * if x\leq \pi /2
    I(x)=\int_{0}^{x}f(t)dt = 1-cos(x)
    * if  x> \pi /2
    \int_{0}^{\pi/2}sin(x)dx + \int_{\pi/2+\delta }^{x}sin(x)dx=1+x-\frac{\pi}{2},\delta  \to \0
    Yes that is correct. I would write as:
    I(x) = \left\{ {\begin{array}{rl}   {1 - \cos (x),} & {x \leqslant \frac{\pi }{2}}  \\   {x + 1 - \frac{\pi }{2}} & {x > \frac{\pi }{2}}  \\ \end{array} } \right.
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  3. #3
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    Re: Riemann integral

    nice! i saw now that i did some typos in my original post, corrected now.
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