In 6.4a we learned about Arc Length and how to find iron a Rectifiable Curve.
In order to do so, you must use this formula:
The “s” represents arc length, and the “a” and “b” are the interval which the length is on.
You simply plug the interval into the formula, as well as the derivative of f(x). Then, you just plug it into your calculator!
Try some practice problems below!
6.4b uses a slightly more complex formula that uses the arc length to find the Surface Area of Revolution.
“If a graph of a rectifiable curve is revolved about a line, the resulting surface is a surface of revolution.”
In order to apply this concept, you must know the following formula:
As you can see, there are two different ways of thinking about this formula.
The first way writes out the entire formula, but the second way simplifies it by just stating it as “radius times the arc length” instead of writing out the arc length formula.
Whichever way works better in your brain is what you should memorize.
Once you have the formula, you plug in both the radius and the arc length (or you plug the derivative of f'(x) into the first formula and do it all at once in your calculator).
Once you put it all in your calculator, calculate it, and round your answer to three decimal places.
Here are a few practice problems to help you out!
Here are some videos to watch if you’re still struggling with the concept!
Don’t worry, sometimes it helps to see it being done, especially for visual learners!
REAL WORLD CONNECTION:
This concept could be used in the real world when trying to find the length of arcs in architecture or finding the distance you have to run around a curved track. There are many times in your life that you deal with arcs, and it could come in handy if you need to know the length or distance you need to travel. You can also apply this to finding the surface area of different objects that are made up of arcs.
Calculus Ants 