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Hello there,

suppose i want to find the arc length of a circle x^2+y^2=R^2 using integration, implicitly differentiating the equation, i find y'=-(x/y)

now,

arc length (circumference)= ([tex]\int[/tex] [tex]\sqrt{1+y'^2}[/tex]dx

putting the value of y'=-(x/y) and substituting for y^2 from the equation of the circle

(y^2= R^2-x^2)

solving, the equation i get

circumference= R*{ sin

where a and b are the limits of integration

whats bugging me here is the limits, when i use the limits [-R,R], i get circumeference=[tex]\pi[/tex]*R

now what i dont get is why do i have to multiply by two to get the actual answer, i mean i didnt use the equation of the upper/lower semicircles ANYWHERE in my calculations, shouldnt these limits be giving me the full circumference, without the need to multiply by two??

suppose i want to find the arc length of a circle x^2+y^2=R^2 using integration, implicitly differentiating the equation, i find y'=-(x/y)

now,

arc length (circumference)= ([tex]\int[/tex] [tex]\sqrt{1+y'^2}[/tex]dx

putting the value of y'=-(x/y) and substituting for y^2 from the equation of the circle

(y^2= R^2-x^2)

solving, the equation i get

circumference= R*{ sin

^{-1}[x/R] }[tex]^{a}_{b}[/tex]where a and b are the limits of integration

whats bugging me here is the limits, when i use the limits [-R,R], i get circumeference=[tex]\pi[/tex]*R

now what i dont get is why do i have to multiply by two to get the actual answer, i mean i didnt use the equation of the upper/lower semicircles ANYWHERE in my calculations, shouldnt these limits be giving me the full circumference, without the need to multiply by two??

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