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Sunday, February 24, 2013

Length - Four Leaf Rose

Category: Integral Calculus, Analytic Geometry, Algebra, Trigonometry

"Published in Suisun City, California, USA"

Find the length of a four leaf rose



Solution:

To draw the given equation in a polar coordinate system, we have to assign the values of θ from 0° to 360° in order to get their values of r as follows

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From the values of r and θ, we can sketch the graph of a four leaf rose as follows


Photo by Math Principles in Everyday Life

The length of a curve is given by the formula





Divide dx and dy by dθ as follows







Next, we need to eliminate dx and dy in the equation above as follows

If 
then

If
then

Therefore,









but 

The above equation becomes



The formula to get the length of any curve at polar coordinate system is



From the given problem that if


then

Since the given curve is symmetrical, we will use one quadrant only in getting the length of a curve and then multiply it later by 4. The range for the angle that we will use is from 0° to 90° or 0 to ½π as follows

















Since the above equation is impossible to integrate by any method, then we have to use the Simpson's Rule in order to integrate the definite integral. Simpson's Rule is more accurate than Trapezoidal Rule and Riemann Sums. The formula for Simpson's Rule is





Let's consider the following, if n = 8, then








Next, let's tabulate the values of f(θ) as follows


Photo by Math Principles in Everyday Life

Therefore







The total length of a Four Leaf Rose is