### 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. 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. It does not want us to evaluate this integral. We recall that for π¦ is equal to some function π of π₯, if π prime is continuous on the closed interval from π to π. Then, we can calculate the length of the arc π¦ is equal to π of π₯ from π₯ is equal to π to π₯ is equal to π by calculating the integral from π to π of the square root of one plus π 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. If we were to set π of π₯ to be equal to sin of π₯, π equal to zero, and π equal to π 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. Then, we would get an integral which represents the length of the curve π¦ is equal to sin π₯ from π₯ is equal to zero to π₯ is equal to π.

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 itβs continuous on the closed interval from zero to π. So our prerequisite that π 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 is equal to the integral from zero to π 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 as the integral from zero to π of the square root of one plus the cos squared of π₯ with respect to π₯.