Length of a curve
Find the length of the curve
y =...

Question

Length of a curve

Find the length of the curve

y = ∫(from 1 to x) sqrt(sqrt(t) - 1) dt, where 1 ≤ x ≤ 16.

Accepted Answer

Welcome back everyone. Find the length of the curve y equals integral from 1 to x of square root of t2 minus 1dt, where x is between 1 and 3 inclusive. A 2 units, b 4 units, c 3 units, and d 9 units. 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 y squad x, so let's begin by focusing on the derivative. We want to identify the derivative of y with respect to x, which is the derivative d divided by dx of the integral from 1 to x of square root of t2 minus 1 dt. What we can do is apply the fundamental theorem of calculus part 1. The lower limit is a constant, and the upper limit is x. So basically we're going to take our function square root of t2 minus 1 and replace t with x, which gives us the square root of x2 minus 1. If we now square the derivative, that would be y2, it is square root of x2 minus 12, and that's x2 minus 1. And what we're going to do is evaluate 1 plus the derivative of y2, which is the part that we have under the radical. That's basically 1 + x2 minus 1. We can conclude that it is x2. And I notice that if we take the square root of x 2, that is equal to the absolute value of x. But since X is between 1 and 3, it is positive, so we can drop the absolute value. And this means that our integral can be written as the integral from 1 up to 3 of x dx. This is an elementary integral, and its value can be found by applying the power rule. We get x to the power of 2 divided by 2 or simply 1/2 x squared. Evaluated from 1 to 3. Applying the fundamental theorem of calculus part two. Previously we applied part one. Right now we are applying part two. We get one half in parent 32 minus 12, and this is equal to 9 minus 1/8 divided by 2. That's 4. So the correct answer is option B. Thank you for watching.

Length of a curveFind the length of the curvey = ∫(from 1 to x) sqrt(sqrt(t) - 1) dt, where 1 ≤ x ≤ 16.

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Length of a curve
Find the length of the curve
y = ∫(from 1 to x) sqrt(sqrt(t) - 1) dt, where 1 ≤ x ≤ 16.