What is the arc length of the polar curve #f(theta) = 5sintheta-4theta # over #theta in [pi/8, pi/3] #?

Answer 1

L #approx# 0.47235219096 #approx# 0.472 (3 decimal places)

The derivation of the arc length of a polar curve over the interval #[a, b]# could be found in another one of my posts, the link to which is here: https://tutor.hix.ai
Applying to this question, we must first find the derivative of the function with respect to #theta#:
#f'(theta) = d/(d theta)(5 sin(theta) - 4 theta)#
Since #d/(d theta) (sin (theta)) = cos(theta)#,
#f'(theta) = 5cos(theta) - 4#
The formula for the arc length of a polar curve over the interval #[a, b]# is given by:
#L = int_a^bsqrt(r^2 + ((dr)/(d theta))^2) d theta#
where #r = f(theta)# and therefore it follows that #(dr)/(d theta) = f'(theta)#

Therefore, the arc length for the given function over the given interval would be:

#L = int_(pi/8)^(pi/3)sqrt((5 sin(theta) - 4 theta)^2 +( 5cos(theta) - 4)^2) d theta#
Using a graphing utility (for the sake of simplicity), the integral equals about #0.47235219096#, or to three decimal places, it is equal to #0.472#
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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