Video Transcript
Write the integral required to calculate the length of the sine curve between 𝑥 is equal to zero and 𝑥 is equal to 𝜋. Do not evaluate it.
The question wants us to write an integral which calculates the length of the sine curve between 𝑥 is equal to zero and 𝑥 is equal to 𝜋. It does not want us to evaluate this integral. We recall that for 𝑦 is equal to some function 𝑓 of 𝑥, if 𝑓 prime is continuous on the closed interval from 𝑎 to 𝑏. Then, we can calculate the length of the arc 𝑦 is equal to 𝑓 of 𝑥 from 𝑥 is equal to 𝑎 to 𝑥 is equal to 𝑏 by calculating the integral from 𝑎 to 𝑏 of the square root of one plus 𝑓 prime of 𝑥 squared with respect to 𝑥.
Since the question wants us to find an integral to express the arc length of the sine curve between 𝑥 is equal to zero and 𝑥 is equal to 𝜋. If we were to set 𝑓 of 𝑥 to be equal to sin of 𝑥, 𝑎 equal to zero, and 𝑏 equal to 𝜋 in our arc length formula. If we could show the derivative of the sin of 𝑥 is continuous on the closed interval from zero to 𝜋. Then, we would get an integral which represents the length of the curve 𝑦 is equal to sin 𝑥 from 𝑥 is equal to zero to 𝑥 is equal to 𝜋.
We know the derivative with respect to 𝑥 of the sin of 𝑥 is equal to the cos of 𝑥. We also know that the cos of 𝑥 is continuous on the real numbers. So in particular, we know it’s continuous on the closed interval from zero to 𝜋. So our prerequisite that 𝑓 prime is continuous on the closed interval between the endpoints of our arc is true.
Therefore, we can conclude the length of the sine curve from 𝑥 is equal to zero to 𝑥 is equal to 𝜋 is equal to the integral from zero to 𝜋 with respect to 𝑥 of the square root of one plus the derivative with respect to 𝑥 of the sine function squared. And we know the derivative of the sine function with respect to 𝑥 is equal to the cos of 𝑥. Therefore, we can represent the length of the sine curve from 𝑥 is equal to zero to 𝑥 is equal to 𝜋 as the integral from zero to 𝜋 of the square root of one plus the cos squared of 𝑥 with respect to 𝑥.
