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Step-by-step explanation:
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Answer:
Step-by-step explanation:
So we have the function:
And we want to find the arc-length from:
By differentiating and substituting into the arc-length formula, we will acquire:
To evaluate, we can use trigonometric substitution. First, notice that:
Let's let y=2x. So:
We also need to rewrite our bounds. So:
So, substitute. Our integral is now:
Let's multiply both sides by 2. So, our length S is:
Now, we can use trigonometric substitution.
Note that this is in the form a²+x². So, we will let:
Substitute 1 for a. So:
Differentiate:
Of course, we also need to change our bounds. So:
Substitute:
The expression within the square root is equivalent to (Pythagorean Identity):
Simplify:
Now, we have to evaluate this integral. To do this, we can use integration by parts. So, let's let u=sec(θ) and dv=sec²(θ). Therefore:
And:
Integration by parts:
Again, let's using the Pythagorean Identity, we can rewrite tan²(θ) as:
Distribute:
Now, let's make the single integral into two integrals. So:
Distribute the negative:
Notice that the integral in the first equation and the second integral in the second equation is the same. In other words, we can add the second integral in the second equation to the integral in the first equation. So:
Divide the second and third equation by 2. So:
Now, evaluate the integral in the second equation. This is a common integral, so I won't integrate it here. Namely, it is:
Therefore, our arc length will be equivalent to:
Divide both sides by 2:
Evaluate:
Evaluate:
Simplify:
Use a calculator:
And we're done!