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Thread: Area between two curves

  1. #1
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    Area between two curves

    For this problem there are two functions
    f(x) = cos x
    g(x) = 2-cos x
    where 0<x<2π
    I know this is probably fairly simple, but we just started learning this stuff
    I know the equation ∫f(x) - g(x) or ∫top-bottom curve, but if you graph it, you can see that it's not that simple (I think)
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  2. #2
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    Quote Originally Posted by Breadordecide View Post
    For this problem there are two functions
    f(x) = cos x
    g(x) = 2-cos x
    where 0<x<2π
    I know this is probably fairly simple, but we just started learning this stuff
    I know the equation ∫f(x) - g(x) or ∫top-bottom curve, but if you graph it, you can see that it's not that simple (I think)
    yes, it's that simple ...

    $\displaystyle A = \int_0^{2\pi} (2-\cos{x}) - \cos{x} \, dx$
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  3. #3
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    Thank you! Yes, I tried that, but the answer I got (plugging into 2x - 2sinx) was 4π, a really big number. It just didn't seem right to me
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  4. #4
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    Quote Originally Posted by Breadordecide View Post
    Thank you! Yes, I tried that, but the answer I got (plugging into 2x - 2sinx) was 4π, a really big number. It just didn't seem right to me
    Not that big a number. When $\displaystyle x= \pi$, cos(x)= -1 so 2- cos(x) is 3 and cos(x)= -1. If you put a rectangle about that figure, it would have height 4 and width $\displaystyle 2\pi$, so area $\displaystyle 8\pi$. It looks reasonable that the area between the two curves is half the area of the rectangle containing them.
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