Calculate the length of the curve? #x = cos(t)+sin(t), y=cos(t)-sin(t)#

#x = cos(t)+sin(t)#
#y=cos(t)-sin(t)#

for #-pi/2 <= t <=pi/2#

1 Answer
Noah G
Mar 1, 2018

We have:

#A = int_(-pi/2)^(pi/2) sqrt(((dx)/(dt))^2 + ((dy)/(dt))^2dt)#

We know that

#(dx)/(dt) = cost - sint#
#(dy)/(dt) = -sint - cost#

#A = int_(-pi/2)^(pi/2) sqrt((cost - sint)^2 + (-sint - cost)^2)dt#

#A = int_(-pi/2)^(pi/2) sqrt(cos^2t + sin^2t - 2costsint + sin^2t + 2sintcost +cos^2t)dt#

#A = int_(-pi/2)^(pi/2) sqrt(2)dt#

#A = [sqrt(2)t]_(-pi/2)^(pi/2)#

#A = sqrt(2)(pi/2) - sqrt(2)(-pi)/2#

#A= (2sqrt(2)pi)/2#

#A = sqrt(2)pi#

The arc length is #sqrt(2)pi# units.

Hopefully this helps!

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