81. Find the lengths of the following curves.
a. y = (x²...

Question

81. Find the lengths of the following curves.

a. y = (x²/8) - ln(x), 4≤x≤8

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Compute the length of the curve. Y is equal to 1 divided by 16 multiplied by x 2 minus 2 multiplied by the natural log of x. 42 less than or equal to x and x is less than or equal to 6. OK. So it appears for this particular problem, we're ultimately asked to determine the length of the curve given our equation for Y and that we're told that 2 is less than or equal to x and x is less than or equal to 6. So now that we know that we're ultimately trying to solve for the length of the curve. Our first step that we need to take is based on our prior knowledge, we need to recall a note that for a function. Y is equal to F of X. We need to note that based on our prior knowledge that the arc length will be from drum roll, X is equal to A. 2, X is equal to B. So let's quickly recap here. So based on our prior knowledge, we need to be able to recall that for a function Y is equal to F of X, that will mean that the arc length from X is equal to A to X is equal to B. Will be written as the following equation L, which represents the arc length, will be equal to the integral evaluated from A to B of the square root of 1 + Dy by dx to the power of 2 dx. So with that said, For our second step is we now need to find DY by DX and we need to find parentheses of DY by DX to the power of 2. Fantastic. So, with that said, we now need to solve for DY. So when we do that, we will find that DY by DX is equal to D by DX 1 divided by 16 multiplied by x 2 minus 2 multiplied by the natural log of x. is equal to 1 divided by 16 multiplied by 2 x minus 2 multiplied by 1 divided by x, which will be equal to x divided by 8 minus 2 divided by x. So now, similarly, we need to solve for a parentheses of Dy by dx to the power of 2. Which will be equal to parentheses of x divided by 8. -2 divided by x in parentheses to the 2, which will be equal to x 2 divided by 16 minus 1 divided by 2 + 4 divided by x to the power of 2. Then that will mean that we could safely write that 1 + parentheses DY so DY by DX to the power of 2. Will be, once again, this will be equal to. One plus. X to the power of 2. Divided by 16 minus minus 1 divided by 2. Plus 4 divided by x to the 2. Which will be equal to x 2 divided by 16 + 1 divided by 2 + 4 divided by x to the power of 2, which will be equal to, when we write it in its most simplified form, it's going to be equal to parentheses x divided by 8. Plus, so once again. X divided by 8 + 2 divided by x in parentheses to the power of 2. Fantastic. So now we need to substitute our variables, or rather our values back into our formula to solve for the arc length. So the length of the curvel is equal to the integral evaluated from 2 to 6 of the square root of parentheses x divided by 8 + 2 divided by x in parentheses to the power of 2. D x, which will mean that l is equal to the integral evaluated from 2 to 6 of parentheses of x divided by 8 plus 2 divided by x d x. Fantastic. Which will mean that L is equal to 1 divided by 8 multiplied by x 2 divided by 2 + 2 multiplied by the natural log of x evaluated from 2 to 6. So essentially we took the once again we're performing our integration. And what we're ultimately trying to do is we're trying to simplify our expression for L down to its most simple form. So right now we're performing our evaluation, and when we perform the evaluation, we will find once we do a little bit of simplifying first, that L will be equal to parentheses of x to the power of 2 divided by 16. Plus 2 multiplied by the natural log of X, and once again we're evaluating from 2 to 6. Fantastic. So, this will mean we can go ahead and write that L is equal to 62 divided by 16 is. Once again, we're plugging in a value of 6 in for x and a value of 2 in for x. So, 62 divided by 16 + 2 multiplied by the natural log of 6 in parentheses minus parentheses of 22 divided by 16 + 2 multiplied by the natural log of 2. And when we simplify this expression down to its most simplified form, skipping a few steps, we will find that L is simply equal to 2 + 2 multiplied by the natural log of 3. So once again, L will be equal to 2 + 2 multiplied by the natural log of 3. And don't forget that our units. For this particular problem will just be units, and that's it. That is our final answer. So usually if they do not give you a specific units to work with, like, for example, they don't tell you if it's centimeters or meters, etc. usually the default units will just be simply units, and you do not want to forget your unit of measurement for your expression. And that's it. We've officially solved for this problem. So going back to the top, we could safely say, based on what we've learned together, that once again our final answer without a doubt will be parentheses 2 + 2 multiplied by the natural log of 3 units, and that's it. That is our final answer for the length of the curve. Thank you so much for watching, hopefully that helped, and I can't wait to see you in the next video. bye.

81. Find the lengths of the following curves.a. y = (x²/8) - ln(x), 4≤x≤8

Watch next

81. Find the lengths of the following curves.
a. y = (x²/8) - ln(x), 4≤x≤8