9–20. Arc length calculations Find the arc length of the follo...

Question

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

y = (x²+2)^3/2 / 3 on [0, 1]

Accepted Answer

Welcome back, everyone. Find the arc length of the curve Y equals 4 X2 + 4 raise to the power of 3 halves divided by 12 on the interval from 0 to 1 inclusive. For this problem, let's recall the RL formula. Our R client L is going to be equal to the integral from 0 to 1 in this case, considering the lower bound in the upper bound of square root of. According to the formula, we take one and we add the square of the derivative of our function. And then we integrate with respect to X. So what we want to do is simply set Y equal to F of X and differentiate F of X. The derivative of F of X would be the derivative of 1/12. We can factor out the constant. 4 X2 + 4 raises the power of three halves. So we're going to take the derivative of our function. That'll be 1/12 multiplied by. First of all, we have a composite function, so we apply the power rule that would be 3 halves multiplied by 4 x 2 + 4, raise to the power of 1/2, and according to the chain rule, we're multiplying by the derivative of the inner function. So we have to multiply by the derivative of 4 X2 + 4. That would be equal to 1/12 multiplied by 3 halves, multiplied by 4 X2 + 4 raises the power of 1/2 multiplied by 8 X, right? That will be the derivative of 4 X2 + 4. And now what we can do is simplify. So we get 1 multiplied by 3 multiplied by 8, which is 24. We also can include X in the numerator. Our denominator contains 12 multiplied by 2, which is 24, so we can immediately cancel out 24. And now we have square root of 4 X2 + 4. We can factor out 4, and we're left with X2 + 1 in. Now, square root of 4 is 2. So we got 2 X square root of X2 + 1. And now what we have to do is square this derivative F of X squared is going to be equal to 4 X2. Multiplied by x2 + 1. Now we are going to add one. To F of X squared. Which is equal to 1 plus let's distribute 4 X to the power of 4 + 4 X squared. What we can notice is that this is a perfect square, right? Specifically, this is equal to 2 X2 + 1 squad. According to the formula, we squared the first term, which gives us 4 X the power of 4, plus 2 multiplied by 1 multiplied by 2 X2, which is 4 X2, plus the square of 1 which is 1, right? So we got a perfect square. So our length is equal to the integral from 0 to 1 of square root of. 2 X2 + 1 squared. And our square root of a square way turns us to solutions, right? They basically cancel each other out. And we're going to take the positive solution because X is between 0 and 1, so our function is going to be positive at those values. Meaning we are integrating from 0 to 1 a function which is 2x2 + 1DX. And how integrating we get. 2, we can factor out the constant, the integral of X where this X cubed divided by 3, and the integral of 1DX is simply X. And then we want to evaluate the result from 0 to 1. So let's apply the fundamental theorem of calculus part 2. We get 2/3 multiplied by 1 cubed. Plus 1, and then we subtract 0, right? Because at the lower limit of 0, our function that we got to execute divided by 3 + X is 0. So we got 2/3 plus 1. And now 2/3 + 1 gives us 5/3. So we got 5/3, since there are no units, we're just going to introduce units. The final answer to this problem is 5/3 units. That would be our our client, and thank you for watching.

9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = (x²+2)^3/2 / 3 on [0, 1]

Key Concepts

Arc Length Formula

Derivative

Integration

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