So, that is my Y axisĪnd this is my X axis right over here. It evaluated at zero, so this is gonna be equal to pi over two minus zero, that's going to be equal to pi over two. We're gonna evaluate that at pi over two and then subtract. One with respect to T is just gonna be T and we're gonna evaluate Well, this is going to be equal to, so you could view it, this is a one here, the antiderivative of Unit circle definition of sin and cosine and so, we have the square root of one, the principle root of one which is just going to be one, so everything here has just simplified to the integral from So, that's one of our mostīasic trig identities, comes straight out of the Squared plus cosine squared of some variable is always Sin times negative sin is positive sin squared, so I could write this as sin squared of T and then DY/DT squared, that's just cosine squared T, plus cosine squared T and then we have our DT out here. Negative's gonna go away, we're gonna have negative That's a negative sin of T squared, well, if you square it the Here is going to be equal to the integral from T is equal to zero to pi over two, that's what we care about, our parameter's going from zero to pi over two of the square root of the derivative of X The derivative of cosine of T is equal to negative sin of T, negative sin of T and what is DY/DT? The derivative of Y with respect to T. Rewritten as this is equal to the integral from A to B of the square root of DX/DT squared plus DY/DT squared DT but either way we can nowĪpply it in this context. Root of the derivative of X with respect to T squared plus the derivative of Y with Point of our parameter, T equals B of the square So, the formula tells us that arc length of a parametric curve, arc length is equal to the integral from our starting point of our parameter, T equals A to our ending We got from the formula actually makes sense. Gonna look at the formula and then we're gonna visualize it and appreciate why what See if you can work that out based on formulas that we It's equal to cosine of T and Y is also defined as a function of T and it's equal to sin of T and we wanna find the arc length of the curve traced out, so length of curve from T is equal to zero to T is equal to pi over two. Say that X is a function of the parameter T and
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