Video: Arc Lengths in a Plane

Write the integral required to calculate the length of the sine curve between π‘₯ = 0 and π‘₯ = πœ‹. Do not evaluate it.

02:07

Video Transcript

Write the integral required to calculate the length of the sine curve between π‘₯ is equal to zero and π‘₯ is equal to πœ‹. Do not evaluate it.

The question wants us to write an integral which calculates the length of the sine curve between π‘₯ is equal to zero and π‘₯ is equal to πœ‹. It does not want us to evaluate this integral. We recall that for 𝑦 is equal to some function 𝑓 of π‘₯, if 𝑓 prime is continuous on the closed interval from π‘Ž to 𝑏. Then, we can calculate the length of the arc 𝑦 is equal to 𝑓 of π‘₯ from π‘₯ is equal to π‘Ž to π‘₯ is equal to 𝑏 by calculating the integral from π‘Ž to 𝑏 of the square root of one plus 𝑓 prime of π‘₯ squared with respect to π‘₯.

Since the question wants us to find an integral to express the arc length of the sine curve between π‘₯ is equal to zero and π‘₯ is equal to πœ‹. If we were to set 𝑓 of π‘₯ to be equal to sin of π‘₯, π‘Ž equal to zero, and 𝑏 equal to πœ‹ in our arc length formula. If we could show the derivative of the sin of π‘₯ is continuous on the closed interval from zero to πœ‹. Then, we would get an integral which represents the length of the curve 𝑦 is equal to sin π‘₯ from π‘₯ is equal to zero to π‘₯ is equal to πœ‹.

We know the derivative with respect to π‘₯ of the sin of π‘₯ is equal to the cos of π‘₯. We also know that the cos of π‘₯ is continuous on the real numbers. So in particular, we know it’s continuous on the closed interval from zero to πœ‹. 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 π‘₯ is equal to zero to π‘₯ is equal to πœ‹ is equal to the integral from zero to πœ‹ with respect to π‘₯ of the square root of one plus the derivative with respect to π‘₯ of the sine function squared. And we know the derivative of the sine function with respect to π‘₯ is equal to the cos of π‘₯. Therefore, we can represent the length of the sine curve from π‘₯ is equal to zero to π‘₯ is equal to πœ‹ as the integral from zero to πœ‹ of the square root of one plus the cos squared of π‘₯ with respect to π‘₯.

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