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

Answer 1
The answer is: #L=1/54(sqrt145-37sqrt37)#.

The curve can be written:

#y=+-4sqrt((x+1)^3)#, but we need only the branch with #y>0#, so:
#y=4sqrt((x+1)^3)#.

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

#L=int_a^bsqrt(1+[f'(x)]^2)dx#.

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

#y=4(x+1)^(3/2)rArry'=4*3/2(x+1)^(1/2)=6sqrt(x+1)#
#L=int_0^3sqrt(1+36(x+1))dx=int_0^3sqrt(36x+37)dx=#
#=1/36int_0^3 36sqrt(36x+37)dx=1/36[(36x+37)^(1/2+1)/(1/2+1)]_0^3=#
#=1/36*2/3[sqrt((36x+37)^3)]_0^3=1/54(sqrt145-sqrt(37^3))=1/54(sqrt145-37sqrt37)#.
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Answer 2

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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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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