# How do you find the length of the curve #y^2 = 16(x+1)^3# where x is between [0,3] and #y>0#?

The curve can be written:

To find the lenght of a curve written as a function and in cartesian coordinates we have to remember this formula:

So, we have first of all calculate the derivative of:

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To find the length of the curve defined by ( y^2 = 16(x + 1)^3 ) where ( x ) is between ( [0, 3] ) and ( y > 0 ), we use the arc length formula for parametric curves:

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

First, we need to express ( x ) and ( y ) in terms of a parameter, say ( t ). We can do this by letting ( y = 4(t + 1)^{3/2} ) and ( x = t ), where ( t ) varies from 0 to 3.

Next, we find the derivatives ( \frac{dx}{dt} ) and ( \frac{dy}{dt} ):

[ \frac{dx}{dt} = 1 ] [ \frac{dy}{dt} = 6(t + 1)^{1/2} ]

Now, we substitute these derivatives into the arc length formula and integrate over the interval [0, 3]:

[ S = \int_{0}^{3} \sqrt{1^2 + \left(6(t + 1)^{1/2}\right)^2} , dt ]

[ S = \int_{0}^{3} \sqrt{1 + 36(t + 1)} , dt ]

[ S = \int_{0}^{3} \sqrt{36t + 37} , dt ]

This integral can be solved using standard integration techniques. Once evaluated, the result will give the length of the curve.

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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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