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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
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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
If
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
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
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| Photo by Math Principles in Everyday Life |
Therefore
The total length of a Four Leaf Rose is
