How to calculate the arc length of a vector function
Formula for finding arc length of a vector function
To find the arc length of the vector function, we will need to use the formula
where is the arc length of the vector function, is the interval that defines the arc, and , , and are the derivatives of the parametric equations of , and respectively.
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To solve for arc length, we’ll need the parametric equations of the vector function. Whether our vector function is given as or , the parametric equations are
Once we have these parametric equations, we’ll take the derivative of each one to get , , and . Assuming we’re given , we’ll have everything we need to use the formula for arc length.
How to calculate the arc length of a vector function over a particular interval
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Finding the arc length of the vector function
Example
Find the arc length of the vector function over the interval .
We’ll pull the parametric equations out of the vector function as
Now we’ll take the derivative of each of these.
To solve for arc length, we’ll need the parametric equations of the vector function.
Plugging the derivatives and the given interval into the formula for arc length, we get
Since , we can simplify the integral to
Evaluating over the interval, we get
The arc length of the vector function over the interval is .