How do you find the arc length of the curve #y=sqrt(x-3)# over the interval [3,10]?

Answer 1

Arc length will be approximately #7.716# units.

Recall that arc length of a curve is given by

#A = int_a^b sqrt(1 + (dy/dx)^2) dx#
The derivative of our curve is given by the chain rule as being #dy/dx= 1/(2sqrt(x -3))#.
#A = int_3^10 sqrt(1 + (1/(2sqrt(x- 3)))^2) dx#
#A = int_3^10 sqrt(1 + 1/(4(x - 3)))dx#
#A = int_3^10 sqrt((4x - 12 + 1)/(4(x -3)))dx#
#A = int_3^10 sqrt((4x - 11)/(4x - 12))dx#

This is a pretty complex integral, so I would solve using a graphing calculator.

Evaluating you should get #A = 7.716# units.

Hopefully this helps!

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Answer 2

To find the arc length of the curve ( y = \sqrt{x - 3} ) over the interval ([3, 10]), follow these steps:

  1. Compute the derivative of the function to find the differential element.
  2. Integrate the square root of (1 + (f'(x))^2) over the given interval.

Starting with the given function ( y = \sqrt{x - 3} ):

  1. Compute the derivative, ( y' ):

[ y' = \frac{1}{2\sqrt{x - 3}} ]

  1. Square the derivative:

[ (y')^2 = \frac{1}{4(x - 3)} ]

  1. Integrate ( \sqrt{1 + (y')^2} ) over the interval ([3, 10]):

[ \text{Arc Length} = \int_{3}^{10} \sqrt{1 + \frac{1}{4(x - 3)}} , dx ]

  1. Evaluate the integral.
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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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