Arc length of a curve in Cartesian coordinates
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Arc length of a curve - in Cartesian coordinates
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The problem under investigation here is to find the length of a curve between two points (say P and Q). The curve is given in terms of a function y = f(x). [Diagram goes here - download the original to see it.] We will designate the required distance s. As with differential calculus, we will arrive at an exact expression for s by using the theory of closer and closer approximations. [Diagram goes here - download the original to see it.] We will approximate the length of the curve by adding up the straight line segments that join points between P and Q. [Diagram goes here - download the original to see it.] As we increase the number of straight line segments the approximation gets better and better. Suppose the length of one of these straight line segments is designated [Equation goes here - download the original to see it.]. Then, as the following diagram indicates: [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] Then the approximate length along the curve is [Equation goes here - download the original to see it.] Hence, the exact curve is [Equation goes here - download the original to see it.] It is the sum of the "infinitesimal" increments along the curve from P to Q. Each "infinitesimal" element is given by: [Equation goes here - download the original to see it.] Also the point P is given by, say, y = f(a) and the point Q by y = f(b). [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] This formula is not in a very manageable form. We are allowed to manipulate on the terms dx and dy as if they were functions, which they are! This enables us to write the arc length in a more convenient form:- [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] This computes the arc length in terms of the variable x¸ but we could also write it in terms of y. [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] We will now illustrate the application of this formula. For example, find the circumference of a circle radius a. [Diagram goes here - download the original to see it.] A circle of radius a has an equation x2 + y2 = a2 this is not a one-one function, which makes integrating it difficult. However, the circle can be treated as four times the length of one of the quadrants. [Diagram goes here - download the original to see it.] On this segment the function is one-one; here we may write:- [Equation goes here - download the original to see it.] Then the arc length of on segment [Diagram goes here - download the original to see it.] is given by [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] This gives the arc length of one quarter of the circumference of the circle. Hence the full circumference is: [Equation goes here - download the original to see it.] The familiar formula which we have now proven. You may not have been aware that [Equation goes here - download the original to see it.]
is a theorem - you have been so familiar with it as a result! logically, the formula for area comes first: [Equation goes here - download the original to see it.] This is not a theorem but actually a definition - the definition of p. [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] Pi is the ratio of the area of a circle of radius r to the area of a square that has a side of length r. This leads us into the question of the properties of Pi, which is a fascinating study. Having defined Pi as we have, we then prove the formula for the circumference, [Equation goes here - download the original to see it.] using the arc length integration we have first shown. Example find the arc length of a parabola x2 = 4y from vertex to x = 2. We have [Equation goes here - download the original to see it.] which is a parabola :- [Diagram goes here - download the original to see it.] The arc length formula is: [Equation goes here - download the original to see it.] The integral of this expression is a standard result found by a hyperbolic substitution and is given by:- [Equation goes here - download the original to see it.] or in logarithmic form:- [Equation goes here - download the original to see it.]
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Contents of
Arc length of a curve in Cartesian coordinates
1 Arc length of a curve - in Cartesian coordinates
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