# How do you find the length of the curve #y=(2x+1)^(3/2), 0<=x<=2#?

Length is 10.38 units. See details below

The length of a curve between a and b values for x is given by

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# 1/27{46sqrt(46)-10sqrt(10)} ~~ 10.384 \ (3dp)#

So then, the arc length is:

And using the power rule for integration, we can integrate to get:

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To find the length of the curve ( y = (2x + 1)^{\frac{3}{2}} ) over the interval ( 0 \leq x \leq 2 ), follow these steps:

- Determine the derivative of ( y ) with respect to ( x ).
- Use the formula for arc length: [ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx ]
- Evaluate the integral over the given interval.

Step 1: Determine the derivative of ( y ) with respect to ( x ): [ \frac{dy}{dx} = \frac{3}{2}(2x + 1)^{\frac{1}{2}} ]

Step 2: Use the formula for arc length: [ L = \int_{0}^{2} \sqrt{1 + \left(\frac{3}{2}(2x + 1)^{\frac{1}{2}}\right)^2} , dx ]

Step 3: Evaluate the integral over the given interval.

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