ARC LENGTH: LOTS AND LOTS OF RIGHT TRIANGLES
click here to enjoy the Arc Length word search
Below is the graph of the function y = 4[x]^.5, divided into intervals points of which are represented by integer values.
An arc is a piece of any curve. Here our arc is a piece of a parabola; by connecting the points we can approximate the length of a curve by adding up the lengths of the line segments.
Each line segment is the hypotenuse of a right triangle and, since we know the coordinates of each point, we can determine each length.
An approximation for the length of this arc would be about 23.6 units.
As we include more and more right triangles to approximate the length:
**the width of each triangle gets closer and closer to zero
**the hypotenuse of the triangle fits a smaller and smaller part of the curve
**the number of right triangles gets closer and closer to infinity
**our approximation of the arc length gets closer and closer to the actual value
DERIVING THE FORMULA FOR THE LENGTH OF AN ARC
Shown above is the function y = [400 - x^2]^.5, which is a circle centered at the origin with a radius of 20 units. If we were to break up the function into 20 intervals, our approximation of the length of this arc should be pretty close.
TESTING OUT THE ARC LENGTH FORMULA
Below is a region of the graph of the function f(x) = [60/x]. Fill in the table values to determine how closely our arc length formula approximates the length of the curve. Draw in the right triangles.
**do the hypotenuses get longer, shorter or stay a constant length?
**how can we check how close our approximation comes to the actual value?
A graphing calculator calculates the length of that arc to be 17.006 units
QUESTIONS TO CONSIDER
- How accurate was your approximation?
- How would you write the summation expression for the arc length you just approximated?
- Did the lengths of the hypotenuses exhibit any discernable pattern? If so, how would you describe the pattern?
let's move on to Series
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