Consider the parametric equation: #x = 15(cos(theta) + (theta)sin(theta))# and #y = 15(sin(theta)  (theta)cos(theta))#, What is the length of the curve for #theta = 0# to #theta = pi/8#?
The Arc Length for a Parametric Curve is given by
So in this problem we have (using the product rule):
So the Arc Length is;
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To find the length of the curve defined by the parametric equations ( x = 15(\cos(\theta) + \theta\sin(\theta)) ) and ( y = 15(\sin(\theta)  \theta\cos(\theta)) ) for ( \theta = 0 ) to ( \theta = \frac{\pi}{8} ), follow these steps:

Compute the derivatives of ( x ) and ( y ) with respect to ( \theta ) to obtain expressions for ( \frac{dx}{d\theta} ) and ( \frac{dy}{d\theta} ).

Use the formula for the arc length of a parametric curve: [ L = \int_{\theta_1}^{\theta_2} \sqrt{\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2} , d\theta ]

Substitute the expressions for ( \frac{dx}{d\theta} ) and ( \frac{dy}{d\theta} ) into the formula.

Integrate the resulting expression from ( \theta = 0 ) to ( \theta = \frac{\pi}{8} ).

Evaluate the integral to find the length of the curve.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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