# How do you find the arc length of #x=2/3(y-1)^(3/2)# between #1<=y<=4#?

Let us evaluate the above definite integral.

I hope that this helps.

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To find the arc length of the curve defined by (x = \frac{2}{3}(y - 1)^{3/2}) between (y = 1) and (y = 4), we use the formula for arc length:

[ L = \int_{y_1}^{y_2} \sqrt{1 + \left(\frac{dx}{dy}\right)^2} , dy ]

First, we find (\frac{dx}{dy}) by differentiating (x) with respect to (y):

[ \frac{dx}{dy} = \frac{d}{dy}\left(\frac{2}{3}(y - 1)^{3/2}\right) ]

[ = \frac{2}{3} \times \frac{3}{2}(y - 1)^{1/2} \times \frac{d}{dy}(y - 1) ]

[ = (y - 1)^{1/2} ]

Now, we substitute (\frac{dx}{dy} = (y - 1)^{1/2}) into the formula for arc length:

[ L = \int_{1}^{4} \sqrt{1 + (y - 1)} , dy ]

[ = \int_{1}^{4} \sqrt{y} , dy ]

[ = \frac{2}{3}y^{3/2} \bigg|_1^4 ]

[ = \frac{2}{3}(4^{3/2} - 1) ]

[ = \frac{2}{3}(8 - 1) ]

[ = \frac{2}{3} \times 7 ]

[ = \frac{14}{3} ]

Therefore, the arc length of the curve between (y = 1) and (y = 4) is (\frac{14}{3}).

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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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