81–88. Arc length Find the arc length of the following curves ...

Question

81–88. Arc length Find the arc length of the following curves on the given interval.

x = sin t, y = t - cos t; 0 ≤ t ≤ π/2

Accepted Answer

Welcome back everyone. Find the arc length of the curve defined by X equals cosine of T and Y equals C plus sine of T or T between 0 and pi divided by 2 inclusive. A 2 square root of 2, B 4 C square root of 2, and D pi. For this problem, let's remember the Rland formula. The Rland L is equal to the integral from A to B of square root of X of T. Squared plus. Y T squared. D C. So what we want to do is identify each derivative. We can begin with X. That's the derivative of cosine of T, which is equal to negative sign of T. Now, Y is the derivative of T + sine of T, which is 1 plus cosine of T. And now let's simplify. We're going to identify. X prime. Squared plus. Y squared. That's negative sign T. Squared Plus 1 plus cosine of T squared. Let's go ahead and square. That would be squared off T plus. One plus. To cosign off T plus. Cosine squared of t. Now, sine squared of T plus cosine squared of T is 1, according to the Pythagorean identity, and then we add another 1, right? So we get 2. Plus To cosine of T. So now our integran becomes square root of 2 in C 1 plus cosine of T, right? We can factor out 2. What we can do is use an identity and in particular one plus cosine of T can be expressed as. 2 cosine squared off t divided by 2 using the half angle formula. And because T is between 0 and 5 divided by 2, when we solve for cosine itself, we're going to have Non-negative values of cosine of T divided by 2, right? Meaning when we take square root, we're going to take the positive root of that. So first of all, we can rewrite our integran as square root of. 2 multiplied by 2, that gives us 4. Cosine squared of T divided by 2. And now when we take square root of that, we take the positive value of cosine based on the domain of T. So we get to cosine of T divided by 2. This is the function that we want to integrate. And now we are ready to set up our integral. The integral is going to go from 0 up to pi divided by 2. And we're integrating to. Cosine of T divided by 2DT. We are also factoring out too because it's a constant. We get 2. The integral of cosine is sin, so we get sine of T divided by 2. And according to the reverse chain rule, we want to divide by One half, right? dividing by 1/2 gives us. Multiplication by 2. So 2 multiplied by 2 gives us an overall coefficient of 4. So our function is 4, sin of T divided by 2, evaluated from 0 up to pi divided by 2. Applying the evaluation theorem, we get 4. In sign off pi divided by 2 divided by 2, that's sign of I divided by 4 minus sign of 0 divided by 2, which is sign of 0. Simplifying, we get 4, sign of pi divided by 4 is square root of 2 divided by 2, and sine of 0 is 0. Now 4 divided by 2 is 2, so our final answer is. 2 square root of 2. Considering the answer choices, we can conclude that the correct answer to this problem corresponds to the answer choice A. Thank you for watching.

81–88. Arc length Find the arc length of the following curves on the given interval.

x = sin t, y = t - cos t; 0 ≤ t ≤ π/2

Key Concepts

Parametric Equations

Arc Length Formula for Parametric Curves

Differentiation of Parametric Functions

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