82. Find a curve through the point (1, 0) whose length from x=...

Question

82. Find a curve through the point (1, 0) whose length from x=1 to x=2 is

L = ∫(from 1 to 2)√(1 + 1/x²)dx.

Accepted Answer

Welcome back everyone. Find the curves passing through the 0 1.2, whose length from x equals 1 to x equals Erl equals the integral from 1 to e of square root of 1 + 9 divided by x 2 dx. For this problem, let's remember the Arkland formula. L is equal to the integral from A to B of square root of 1 plus the derivative of y2 d x. And in this problem we have the integral from 1 to e of square root of 1 + 9 divided by x 2 d x. So by analogy we can notice that. The derivative of y2 is going to be equal to 9 divided by x2. What we're going to do is solve for the derivative by taking the square root of each side, so y is going to be positive or negative, square root of 9 divided by x2, which is positive or negative 3 divided by x. Now we have our derivative. We're going to identify Y itself. Y is going to be the integral of the derivative, right, so we want to integrate positive or -3 divided by x dx. We can factor out the constant which is 3, and we get positive or -3 integral of d x divided by x, and this is an elementary integral. So we get positive or negative 3 lm of the absolute value of x, right, plus a constant of integration c. Now, additionally, since the limits of integration are from 1 toe, we can basically drop the absolute value. And now our goal is to identify the constants of integration C. Let's go ahead and start with the positive solution, right? So that we Ln of considering the point we have x equals 1 y equals 2, right, so we're going to take 3 ln off 1 plus c is going to be equal 2 and from here we will identify the constant c. Now LN of 1 is equal to 0, so we get C equals 2, and this is how we get our first. Curve right y equals. 3 ln of x + 2. Or the negative solution. We're going to take -3 LM. Of one Plus c equals 2 again using the same point. We know that. LN 1 is equal to 0, so again we get C equals 2. And our next curve is y equals -3 ln x + 2. Well then, this is how we get our curves. Thank you for watching.

82. Find a curve through the point (1, 0) whose length from x=1 to x=2 isL = ∫(from 1 to 2)√(1 + 1/x²)dx.

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82. Find a curve through the point (1, 0) whose length from x=1 to x=2 is
L = ∫(from 1 to 2)√(1 + 1/x²)dx.