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Problem 12 Easy Difficulty

Find the are length parameter along the curve from the point where $t=0$ by evaluating the integral
$s(t)=\int_{0}^{t}|\mathbf{v}(\tau)| d \tau$
from Equation (3). Then use the formula for $s(t)$ to find the length of the indicated portion of the curve.
$$\mathbf{r}(t)=(\cos t+t \sin t) \mathbf{i}+(\sin t-t \cos t) \mathbf{j}, \quad \pi / 2 \leq t \leq \pi$$

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Video Transcript

we have given function and we're us to find its arc length perimeter and use our to find the length in the given interrupt. So to find the are killing perimeter, we just take the integral from zero to t off the speed tell detail and then the speed. I just noted it in terms O b. We got ahead. So until find this bait, we take two driven it, which gives us diversity. So I'm going to put that in terms of this we had. And then we take the derivative, always competent, so called science. He gives us negative, scientific and then this is a product ruling here. So it will be one time scientific. So posts I in team on plus t co Sainty off I and then same pink over here. The drill it off Sign each just because saying and then over here is just negative born times cause I ain't used to read native Coursing key and then plus T Santee o J the's turns cancel out and this castle out So our velocity are our prime of t is equal to t cause I know t i plus t sign up t j now to find the magnitude speed. We just take his distance for Melo magnitude. Still, our speed is equal to square it of T square co sane square key plus t square ST square of tea. And then we can pack throughout T Square and then it gives us a trick. I'd any off course and square plus n square, which is one. So we have root two square and decisions t now, to find out our clients permit er will just used a formula. So are massive. T will be from Sierra to t A now, just not a beastie is town detail. So decisions is how squared ability to problem zero tasker over to you really do from zero to t. So what we have in here is just he squared over to so are a safety is equal to keen square over two and now we just need to plug in the values off the given interval. So our us off my house old is equal to high Hafs Square times 1/2 it gives us hi square off eight and then we'll use pie. So are a self pie. It's just hi squared over two, so and then to get the actual are Klink will disturb Proact to get the interval. This does correctly. And then our arc length are us well equal to hi. Square over two minus pi squared over. Eat And then we bring you to common denominator Real Ghadry pi square over a And this is our answer for the second part of the question.

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