Finding arc length
Find the length of the curve
y ...

Question

Finding arc length

Find the length of the curve

y = ∫ from 0 to x of √(cos(2t)) dt, 0 ≤ x ≤ π/4.

Accepted Answer

Find the length of the curve, Y equals the integral from 0 to x square root sin 2 TDT for X between 0 and pi divided by 2. We have 4 possible answers, being 341, or 2, all in the units. Now to find the length of a curve, we will use the arcfleet formula. L equals The integral from A to B of the square root 1 + Dyd x 2. DX Now, to solve this, we first need to find DYDX, which is the derivative. So with the DY and DX of Y. In our case, Y is the integral from 0 to X square root sin to TDT. Which means DYDX is going to be the derivative. Of our integral from 0 to x of square root sin to TDT. Now, based on the fundamental theorem of calculus, part one, The derivative of an integral with respect to its upper limit is simply the integral evaluated at that limit. In other words, we will replace our T with our upper limit of X. This means DYDX is just the square root sin to X. We cannot substitute this into our arc length formula. L equals the integral. From 0 to pi divided by 2. Of The square root 1 + square root sin 2 x. All squared d x. This simplifies even further to 0 to pi divided by 2 of the square root, 1 plus sin 2 x dx. Now, to solve this, we can use trig identities to expand our portion underneath the radical. Now we know that one is equals. To sin square x plus cosine square x. We also know sin 2 x equals to 2 sin x cosine x. Meaning we can then rewrite our integral. From 0 to pi divided by 2, as the square root of sin square x plus cosine square x, plus 2 sin x cosine x. With dx at the end. Now we notice then that we have. A trinomial, which can be factored. We can actually factor this to be the integral from 0 to pi divided by 2 of the square root. Of Sin x plus cosine x all squared. Now we have the square root of this. Which means We have the integral from 0 pi divided by 2 of sin x plus cosine x dx. And this will be inside the absolute value. But because we have the interval from 0 to pi divided by 2, and both sine and cosine are non-negative, we don't need these absolute value bars. Now we can just evaluate our integral. We have the integral of sine and cosine, which gives us negative cosine x plus sin x. From 0 to pi divided by 2. We can now evaluate. We have negative cosine pi divided by 2 plus sine, pi divided by 2 minus negative cosine of 0 plus sine of 0. We do know the answers to every one of these based on the unit circle. We get 0. Plus 1 Minus -1 + 0. Which is just 1 + 1 or 2. And this is in units. So, our answer of 2 units corresponds to answer D. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Finding arc lengthFind the length of the curvey = ∫ from 0 to x of √(cos(2t)) dt, 0 ≤ x ≤ π/4.

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Finding arc length
Find the length of the curve
y = ∫ from 0 to x of √(cos(2t)) dt, 0 ≤ x ≤ π/4.