# Consider the parametric equation #x = 9(cost+tsint)# and #y = 9(sint-tcost)#, What is the length of the curve for #t= 0# to #t=3pi/10#?

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To find the length of the curve defined by the parametric equations ( x = 9(\cos t + t \sin t) ) and ( y = 9(\sin t - t \cos t) ) for ( t = 0 ) to ( t = \frac{3\pi}{10} ), we can use the arc length formula for parametric curves:

[ L = \int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt ]

where ( t_1 = 0 ) and ( t_2 = \frac{3\pi}{10} ).

[ \frac{dx}{dt} = -9t \sin t ] [ \frac{dy}{dt} = 9t \cos t ]

Substituting these derivatives into the arc length formula:

[ L = \int_{0}^{\frac{3\pi}{10}} \sqrt{\left(-9t \sin t\right)^2 + \left(9t \cos t\right)^2} dt ] [ L = \int_{0}^{\frac{3\pi}{10}} \sqrt{81t^2 \sin^2 t + 81t^2 \cos^2 t} dt ] [ L = \int_{0}^{\frac{3\pi}{10}} \sqrt{81t^2(\sin^2 t + \cos^2 t)} dt ] [ L = \int_{0}^{\frac{3\pi}{10}} 9t , dt ] [ L = \left[\frac{9t^2}{2}\right]_{0}^{\frac{3\pi}{10}} ] [ L = \frac{81\pi^2}{200} ]

Therefore, the length of the curve for ( t = 0 ) to ( t = \frac{3\pi}{10} ) is ( \frac{81\pi^2}{200} ).

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To find the length of the curve defined by the parametric equations (x = 9(\cos t + t \sin t)) and (y = 9(\sin t - t \cos t)) for (t = 0) to (t = \frac{3\pi}{10}), we can use the formula for the arc length of a parametric curve:

[ L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} , dt ]

First, we need to find the derivatives of (x) and (y) with respect to (t):

[ \frac{dx}{dt} = 9(-\sin t + t \cos t + \cos t - t \sin t) ]

[ \frac{dy}{dt} = 9(\cos t - t \sin t + \sin t + t \cos t) ]

Next, we plug these derivatives into the formula:

[ L = \int_{0}^{\frac{3\pi}{10}} \sqrt{\left(9(-\sin t + t \cos t + \cos t - t \sin t)\right)^2 + \left(9(\cos t - t \sin t + \sin t + t \cos t)\right)^2} , dt ]

[ L = \int_{0}^{\frac{3\pi}{10}} \sqrt{81(1 + t^2)} , dt ]

[ L = 9 \int_{0}^{\frac{3\pi}{10}} \sqrt{1 + t^2} , dt ]

We recognize this as the integral form of the arc length of the curve (y = \sqrt{1 + x^2}), which is (L = \frac{1}{2}(e^{2\pi}-1)). Thus, the length of the curve for (t = 0) to (t = \frac{3\pi}{10}) is approximately (9(\frac{1}{2}(e^{2\pi}-1))).

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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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