81–88. Arc length Find the arc length of the following curves ...

Question

81–88. Arc length Find the arc length of the following curves on the given interval.

x = 2t sin t - t² cos t, y = 2t cos t + t² sin t; 0 ≤ t ≤ π

Accepted Answer

Hello. In this video, we are going to be finding the arc length of the given parametric curves on the interval from 0 to pi. Now in order to find the arc length of a given pair of parametric curves, that arc length is defined as l equal to the integral from A to B of the square root of the derivative with respect to t, derivative of x with respect to t squared, plus the derivative of y with respect to t squad. So, what we need to do is we need to find the derivative of our X parametric curve, and the derivative of our Y parametric curve first. Now, X is equal to 3 multiplied by 2 tint minus T2 cosine of T. Now the derivative is going to require us to use the power rule twice, but by taking the derivative of x, we get d x dt is equal to 3 multiplied by the quantity 2 + t squared. Multiply by sin of t. Now, the same thing can be said about Y. Y is equal to 3 multiplied by 2T cosine of T + T 2 sin of T. We will have to take the derivative by using the product rule twice, but the derivative is going to be DYDT equal to 3 multiplied by the quantity, 2 + T 2 multiplied by cosine of T. So these are going to be the derivatives that we are going to plug into the given integral. But notice that we have to square both of the derivatives next, so we are going to square both of the derivatives d x dt squared gives us the quantity. 9 Multiplied by 2 + t quantity squared. Or 2 + T 2 quantity squared multiplied by sin squared of t. And squaring DY gives us DYDT quantity squad equal to 9 multiplied by 2 + T 2 quantity squad, cosine squared of T. Next, what we're going to do is we're going to take the sum of both of these squared derivatives. That is going to give us DXDT quantity square plus DY DT quantity squared, and that is going to simplify to give us 9 multiplied by 2 + T 2 quantity squared multiplied by sin squaret plus cosine squaredt. This will further simplify to be 9 multiplied by 2 + t2 quantity squared. And finally, we are going to take the square root in order to simplify the integrand. The square root. Of D X D T quantity squared. Plus Dydt quantity squared is equal to the square root of 9 multiplied by 2 + t square quantity squared, and this is going to further simplify to be 3 multiplied by the quantity 2 + t squared. So this is going to be the fully simplified version of the integrand, and keep in mind that we are integrating on the boundary from 0 to pi. So our arc length integral is going to be defined as the integral from 0 to pi of 3 multiplied by 2 + T 2 DT. So, we will integrate term by term, that is going to leave us with 3 multiplied by 2 T plus 1/3 T cubed, and this will be evaluated from 0 to pi. Finally, this will simplify to be 3 multiplied by 2 pi plus 1/3 multiplied by pi cubed. And by distributing the 3, we are left with 6 pi plus pi cubed, which is going to be the final solution to the problem. So this is going to be the arc length of the given parametric equation, and with that being said, the solution to this problem is going to be c. So I hope this video helps you in understanding how to find arc length, and we will go ahead and see you all in the next video.

81–88. Arc length Find the arc length of the following curves on the given interval.x = 2t sin t - t² cos t, y = 2t cos t + t² sin t; 0 ≤ t ≤ π

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81–88. Arc length Find the arc length of the following curves on the given interval.

x = 2t sin t - t² cos t, y = 2t cos t + t² sin t; 0 ≤ t ≤ π