Trigonometry Arc Length Problem

[Euler2718]

First I'd just like to point out that I'm taking calculus and advance pre-calculus simultaneously (kind of a stupid system) and this is a problem in the pre-calc.

1. Homework Statement

2. Homework Equations

Let 'a' be arc length.

$$a=\theta r$$

$$a = \int_{a}^{b} \sqrt{1+[f'(x)]^{2}} dx$$

3. The Attempt at a Solution

The lecture on arc length wasn't even a lecture. Merely a hand out of the chapter summery, not to be touched again. I saw no central angle so $$a=\theta r$$ seems to be out of the question. This seemed to be an unusual problem, so I used Pythagoras after a while. I went

$$( \Delta x )^{2} + (\Delta y)^{2} = a^{2}$$

This yields the arc length to $$\sqrt{8}$$ units.

I confirmed this with the integration for arc length. But I'm really not at the level of arc length integration yet. I can do basic integration only (I skipped ahead in the course a bit). Via wolfram alpha:

$$\int_{-8}^{-6} 10\sqrt{\frac{1}{100-x^{2}}} dx = 10\sin ^{-1}\left(\frac{7}{25}\right) \approx 2.83794$$

Which is pretty much $$\sqrt{8}$$ So really, my question is if there is another more formal way to approach this problem using trigonometric concepts to arrive at an answer.

Thank you in advance.

 

[phinds]

Can you figure out the angle AQB?

 

[Euler2718]

Can you figure out the angle AQB?

Ahhh. Using the right triangles no doubt. I'll give it a go. Thanks.

 

[Mark44]

Problems involving derivatives or integrals should be posted in the calculus section, not in the precalculus section. I have moved this thread.

 

[Euler2718]

Problems involving derivatives or integrals should be posted in the calculus section, not in the precalculus section. I have moved this thread.

Sorry. I merely included the calculus to show what I used to compute the answer, as it was technically part of my attempt.

 

[Euler2718]

Pretty sure I got it now! Thank you phinds for the tip. I calculated the central angle to be 16.3°, and used the arc length equation from there. It comes out to 2.8 units, which is what the Pythagorean approximation and the integration came to.

 

[phinds]

Sounds good.