41–48. Geometry problems Use a table of integrals to solve the...

Question

41–48. Geometry problems Use a table of integrals to solve the following problems.

42. Find the length of the curve y = x^(3/2) + 8 on the interval from 0 to 2.

Accepted Answer

Welcome back everyone. In this problem, determine the length of the curve Y equals X to the core of 3 halves plus 3 from X equals 2 to X equals 5. Now we're trying to find the length of the curve, the arc length. What do we know about a function's arc length? Well, recall, OK, that for a function Y equals F of X, OK. The airle And the arc lens. From X equals A, OK. So X equals B. Is going to be equal to and let's call that arc length l the integral between the bones A and B of the square root of 1 plus the derivative of Y with respect the X squared with respect to x. So what this is telling us is that for our function Y, which in this case is equal to X to the power of 3 halves plus 3, if we can differentiate Y with respect to X, we can substitute it into this formula for the arc length to integrate and solve 4 L. Now, here, we know that, let me write it here. Y is equal to x of 3 halves plus 3, OK. So that means that the derivative of Y with respect to X will be equal to 3 halves of X to the power of 1/2. We also know that a or lower boundary is equal to 2 and B or upper boundary is equal to 5. So substituting all of this into the Arle formula, that means the length is going to be equal to the integral between 2 and 5 of the square root of 1 + 3 halves of X to the pore of a half squared with respect to x. And when we simplify that, that's equal to the integral between 2 and 5 of the square root of 1 + 9/4 of X, or we could write that as 9/4 of X plus 1 with respect to X. Now, let's integrate this by substitution. To do that, let's let you be equal to 9/4 of X plus 1. Well, in that case, the derivative of u with respect to X is going to be equal to 9/4, which means that d x. Is going to be equal to 4/9 of the U. Now, if we also change the boundaries or the bounds, when X is equal to 2, OK, then you will be equal to 9/4 of 2, OK, plus 1. And that is going to be equal to 11 halves. On the other hand, When x equals 5, you will be equal to 9/4 of 5 plus 1, which is going to be equal to 494. So know that we've changed the bones, we've expressed DX in terms of you and X and you in terms of X. Then we can go ahead and substitute in our formula, OK, in our integral. So now this tells us that L then is going to be equal to the integral between the boundaries of 11 halves and 49/4 of the square root of U multiplied by 4/9 of DU. No, we could rewrite that as 4 9th software definite integral, yeah. Are of the definite integra between our bones for you to the poor of a half with respect to you, and when we integrate, that's gonna give us 4 9s multiplied by 2/3 of you to the poor of three halves, OK. Between our bones of 11 halves and 494s. All you need to do now is to simplify and solve. So now when we go ahead here, we're going to get this to be 49s multiplied by 2/3, 8/27 multiplied by 49/4. To the core of three halves. 11 halves to the power of 3 halves. I know when we continue here, we, we, we expand our, our power or we apply our power. So that's gonna be 827 multiplied by 343 divided by 8, OK, minus 11 route 11 divided by 2 route 2. Now if we try to combine these by writing them with the same denominator, we take it a step further. This is going to be 827 multiplied by 343 minus 22 root 2 divided by 8. And when we cancel it here in our expression. Then this is going to give us an arc length L of 343 minus 22 route 22 divided oops, this should have been route 22, my apologies. So 22 route 22 divided by 27. Thus this is our arc length L. Thanks a lot for watching everyone. I hope this video helped.

41–48. Geometry problems Use a table of integrals to solve the following problems.
42. Find the length of the curve y = x^(3/2) + 8 on the interval from 0 to 2.

Key Concepts

Arc Length Formula

Differentiation

Definite Integrals

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