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Sum of area bounded by the curve

  1. Dec 29, 2015 #1
    Why we sometimes take the area bounded by the curve is sum of positive area and absolute of negative area(e.g. ∫\int_0^2π sin(x)\, dx is equal to 4 or area of ellipse )?But sometimes we just sum positive and negative areas which is equal to 0(e.g. area of cycloid →when we integrate we get r*r(x-2sin(x)+½x+sin(2x)/4) from 0 to 2π and we get r*r(2π-2*0+π+0)-0=3r*r*π).Why I can´t just take for -2sin(x) from 0 to 2π is equal to 4 and equally for sin(2x)/4,because that is the area bounded by the curve?
     
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  3. Dec 29, 2015 #2
    An integral like $$ \int_a^b f(x)\,dx $$ calculated as the signed area between ##f(x)## and ##x## axis.
    If there are conditions that split this integral to partial integrals in ##[a,b]##, then may have different sign.
     
  4. Dec 30, 2015 #3
    Is then area of
    \int_0^2π sin(x)\,dx

    0 or 4?If we take

    \int_0^2π sin(x)\,dx = 2 \int_0^π sin(x)\,dx

    then area be 4(what is correct because that is area bounded by the curve).I can´t understand how area in this case could be 0 because area is not a vector.Thanks.
     
  5. Dec 30, 2015 #4
    $$\int_0^{2π} sin(x)\,dx = 2 \int_0^π sin(x)\,dx$$ may used on cases than something change on ##\pi## value. Generally is zero.
    Please, place the foul problem.
     
  6. Dec 30, 2015 #5

    Samy_A

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    That's not correct.
    LHS is 0, RHS is 4.
     
  7. Dec 30, 2015 #6
    Yes, is not correct but when we don't explain what we do, some people thinks that may be correct for some reason.
    For example, I don't know what is LHS or RHS. Is something like KGB to me.
     
  8. Dec 30, 2015 #7

    Samy_A

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    Better having problems with LHS or RHS than with the KGB. :)
    LHS is shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side.
     
  9. Dec 30, 2015 #8
    That´s my problem, why we in complex integrals with sin(x) (from 0 to 2 π) use LHS for sin(x)?But when we calculate area bounded only with sinusoid (by 0 to 2π) ,we use RHS.
     
  10. Dec 30, 2015 #9
    What does "area" exactly mean? Area for me is the paper amount that need painting and for this reason I take right hand result. But the integral have sign and is the left hand result.
     
  11. Dec 30, 2015 #10

    Samy_A

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    It depends on the question.
    The integral of the sine function between 0 and 2π is the LHS, with result 0.
    But if you are asked to compute the area between the sine function and the x-axis, for x between 0 and 2π (say for a real estate project), then you have to add the "positive" area between 0 and π, and the "negative" area between π and 2π. In this case the RHS is what you want.
     
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