# What is the arclength of #f(x)=cos^2x-x^2 # in the interval #[0,pi/3]#?

arc length

(Note: Used TI-84 calculator to integrate)

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To find the arc length of the function f(x) = cos^2(x) - x^2 on the interval [0, π/3], you can use the arc length formula:

Arc length = ∫√[1 + (f'(x))^2] dx, where f'(x) is the derivative of f(x).

First, find the derivative of f(x):

f'(x) = -2x + 2cos(x)(-sin(x))

Then, compute (f'(x))^2:

(f'(x))^2 = (-2x + 2cos(x)(-sin(x)))^2

Now, substitute f'(x) and (f'(x))^2 into the arc length formula and integrate over the interval [0, π/3]:

Arc length = ∫[0 to π/3] √[1 + (-2x + 2cos(x)(-sin(x)))^2] dx

Evaluate this integral to find the arc length.

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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.

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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