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Help with Area Question

  1. May 3, 2014 #1
    1. The problem statement, all variables and given/known data

    Problem is attached in this post.

    2. Relevant equations

    Problem is attached in this post.

    3. The attempt at a solution

    I set my Integral as ∫cos2π dx from 0 to 1/2 and get an answer of 0, which is incorrect.

    The correct answer is 1/π, but I don't understand why etc.
     

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  2. jcsd
  3. May 3, 2014 #2

    Dick

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    If you mean the integral of cos(2πx) from x=0 to x=1/2, then I think the answer is 0. But part of that area is above the x-axis and part is below. I think they want you to add them. Just doing the integral will subtract them.
     
    Last edited: May 3, 2014
  4. May 4, 2014 #3
    How exactly would I add them, what method should I use etc.?
     
  5. May 4, 2014 #4

    SammyS

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    Graph the function from x= 0 to x=1/2, and see what Dick means.
     
  6. May 4, 2014 #5

    LCKurtz

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    @student93: Remember that ##\int_a^bf(x)~dx## is the area under the graph of ##y=f(x)## if ##f(x)\ge 0##. The more general form is $$
    \text{Area} = \int_a^b y_{upper} - y_{lower}~dx$$The first formula above works because ##y_{upper} = f(x)## and ##y_{lower} = 0## when you are finding the area under a nonnegative function and above the ##x## axis. Do you see why ##\int_0^{\frac 1 2} \cos(2\pi x)~dx## isn't in the form ##y_{upper}-y_{lower}## in the integrand?
     
  7. May 4, 2014 #6
    Yes, but then how do I find the area? I'm having trouble setting up an integral that works etc.
     
  8. May 4, 2014 #7

    Dick

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    In the interval [0,1/2], where is cos(2*pi*x) positive and where is it negative? Integrate the regions separately and then think how to combine the numbers.
     
  9. May 4, 2014 #8

    ehild

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    The shape is bounded also by the x axis. See picture. Imagine you have to paint the blue area on a wall. You need 1 litre paint to 1 m^2. Calculate how much paint you need. Can you use negative amount of paint?

    ehild
     

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  10. May 4, 2014 #9

    LCKurtz

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    Look at ehild's picture in post #8. Notice that ##y_{upper}## and ##y_{lower}## are two piece functions. Set up two integrals accordingly.
     
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