9–20. Arc length calculations Find the arc length of the follo...

Question

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

x = 2e^√2y + 1/16e^−√2y, for 0 ≤ y ≤ ln²/√2

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says determine the arc length of the function X, which is equal to 3. Multiplied by E rates to the quantity of the square to 3 multiplied by Y in quantity, plus 1 divided by 36, multiplied by E rate to the quantity of minus the square root 3 multiplied by Y in quantity or the interval 0 is less than or equal toy, which is less than or equal to the square root 3 multiplied by the natural log of 3 divided by 3. So we're asked to find the arc length of our function, and here we need to recall our formula for finding the arc length. So our arc length L is going to be equal to the integral from A to B of the square root of quantity of 1 plus the quantity of the derivative of X with respect to Y in quantity squared in quantity D Y. So the first step is to actually calculate the derivative of X with respect to Y, and this is going to be equal to the derivative with respect toy of the quantity of 3 multiplied by E raised to the quantity of the square root of 3, multiplied by Y in quantity, plus 1 divided by 36, multiplied by E raised to the quantity of minus square root of 3 multiplied by Y in quantity. So taking this derivative, this is going to be equal to 3, multiplied by the square root of 3, multiplied by E rates to the quantity of the square root of 3, multiplied by Y in quantity. Minus the square root of 3, divided by 36, multiplied by E rates to the quantity of minus the square root of 3, multiplied by Y in quantity. So next we'll actually want to square this result, since we're really interested in the quantity of the derivative of X with respect to Y in quantity squared. And so this is going to be equal to the quantity of 3, multiplied by the square root of 3, multiplied by E, raised to the quantity of the square root of 3, multiplied by Y in quantity minus. The square root of 3, divided by 36, multiplied by E rates to the quantity of minus the square root of 3, multiplied by Y in quantity, then another in quantity, and then squared. So multiplying everything out, this is going to be equal to. 27 multiplied by E raises to the quantity of 2 multiplied by the square root of 3, multiplied by Y in quantity. -1 divided by 2. Plus 3 divided by 36 squared, multiplied by E raised to the quantity of minus 2 multiplied by the square root of 3, multiplied by Y in quantity. Now, if we look at our integram, we see that underneath the radical, we have the quantity of 1, plus the quantity of the derivative of X with respect to Y in quantity squared. And so this is going to be equal to 1 plus. 27 multiplied by E rates to the quantity of 2 multiplied by the square root of 3 multiplied by Y and quantity minus 1/2 plus 3 divided by the quantity of 36 in quantity squared multiplied by E raised to the quantity of minus 2 multiplied by the square root of 3, multiplied by Y in quantity. Which way to simplify this is going to be equal to 27 multiplied by E raised to the quantity of 2 multiplied by the square root of 3, multiplied by Y in quantity, plus 12, plus 3 divided by 36 squad, multiplied by E raises the quantity of minus 2 multiplied by the square root of 3, multiplied by Y in quantity. But this is actually equal to the quantity of 3 multiplied by the square root of 3. Multiplied by E raised to the quantity of the square root of 3, multiplied by Y and quantity. Plus the square root of 3 divided by 36, multiplied by E raised to the quantity of minus square root of 3, multiplied by Y in quantity, then another in quantity and then squared. So, for our integrand, for our arc length, we'll just need to take the square root of this. So that means that our arc length L. is going to be equal to the integral and our limits of integration are going to go from 0 to the square root of 3, multiplied by the natural log of 3, divided by 3. Of the quantity of 3, multiplied by the square root of 3, multiplied by E, rates of the quantity of the square root of 3, multiplied by Y. Plus, the square root of 3, divided by 36, multiplied by E raised to the quantity of minus the square root of 3, multiplied by Y in quantity. And that gets multiplied by DY. And so now we just need to perform the integration, and we're gonna integrate term by term. So, our length L is going to be equal to. 3, multiplied by E raised to the quantity of the square root of 3, multiplied by Y in quantity. Plus, The quantity of minus 1 divided by 36 in quantity multiplied by E raised to the quantity of minus 3 multiplied by Y in quantity. And this whole quantity needs to be evaluated from 0 to the square root of 3, multiplied by the natural log of 3 divided by 3. So substituting in our limits of integration, this is going to be equal to. 3 multiplied by E raised to the quantity of the square root of 3, multiplied by the quantity of the square root of 3, multiplied by the natural log of 3, divided by 3 in quantity. Then we'll have minus 1 divided by 36, multiplied by E raised to the quantity of minus the square root of 3, multiplied by the quantity of the square root of 3, multiplied by the natural log of 3, divided by 3. In quantity, then we'll have minus. 3 multiplied by E rates to the quantity of the square root of 3 multiplied by 0 in quantity. Plus 1 divided by 36. Multiplied by E raised to the quantity of minus the square root of 3, multiplied by 0 in quantity. So now we just need to simplify this expression. In doing so, this is going to be equal to 3, multiplied by E raised to the quantity of The natural log of 3. Then we'll have minus 1 divided by 36, multiplied by E raised to the quantity of minus the natural log of 3 in quantity. You'll have -3, multiplied by E to 0. Plus, 1 divided by 36, multiplied by E to the 0 power. And so we'll find further. This is going to be equal to. 9 Minus 1 divided by 108. -3. Plus 1 divided by 36. And so continuing to simplify, this is going to be equal to. 6 Plus, 1 divided by 54, and combining these two values, this is going to be equal to. 325 divided by 54 units. And so this here is the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

9–20. Arc length calculations Find the arc length of the following curves on the given interval.x = 2e^√2y + 1/16e^−√2y, for 0 ≤ y ≤ ln²/√2

Watch next

9–20. Arc length calculations Find the arc length of the following curves on the given interval.
x = 2e^√2y + 1/16e^−√2y, for 0 ≤ y ≤ ln²/√2