How do you find the arc length of the curve #f(x)=2(x-1)^(3/2)# over the interval [1,5]?

Answer 1

Use the arc length formula.

#f(x)=2(x−1)^(3/2)#
#f'(x)=3sqrt(x-1)#

Arc length is given by:

#L=int_1^5sqrt(1+9(x-1))dx#

Simplify:

#L=int_1^5sqrt(9x-8)dx#

Integrate directly:

#L=2/27[(9x-8)^(3/2)]_1^5#

Hence:

#L=2/27(37^(3/2)-1)#
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Answer 2

To find the arc length of the curve ( f(x) = 2(x-1)^{\frac{3}{2}} ) over the interval ([1, 5]), you can use the arc length formula:

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

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

[ f'(x) = \frac{d}{dx} [2(x-1)^{\frac{3}{2}}] ]

Then apply the arc length formula with the limits of integration from 1 to 5. Calculate the integral, and that will give you the arc length of the curve over the interval ([1, 5]).

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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.

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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