Find the arc length for the following question?

enter image source here

1 Answer
May 7, 2018

2

Explanation:

For x = x(y), arc length is s = int_(y_1)^(y_2) \ sqrt(1 + (x')^2) \ dy

From Fundamental Theorem of Calculus:

  • x(y) = int_1^y \ sqrt (sec^4 t - 1 ) \ dt implies x' = sqrt (sec^4 y - 1 )

So:

s = int_(- pi/4 )^(pi/4) \ sqrt(1 + (sqrt (sec^4 y - 1 ))^2) \ dy

= int_(- pi/4 )^(pi/4) \ sec^2 y \ dy

= int_(- pi/4 )^(pi/4) \ (tan y)^' \ dy

= ( \ tan y )_(- pi/4 )^(pi/4)= 2