# What is the arclength of #f(x)=x-sqrt(x+3)# on #x in [1,3]#?

Arclength is given by:

Simplify:

Complete the square in the square root:

This is a known integral. If you do not have it memorized apply integration by parts or look it up in a table of integrals:

Reverse the substitution:

Insert the limits of integration:

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To find the arc length of ( f(x) = x - \sqrt{x + 3} ) on the interval ( [1, 3] ), we use the formula for arc length:

[ L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} , dx ]

where ( f'(x) ) is the derivative of ( f(x) ).

First, find ( f'(x) ):

[ f'(x) = 1 - \frac{1}{2\sqrt{x + 3}} ]

Now, plug ( f'(x) ) into the arc length formula and integrate over the interval ( [1, 3] ):

[ L = \int_{1}^{3} \sqrt{1 + \left(1 - \frac{1}{2\sqrt{x + 3}}\right)^2} , dx ]

This integral may require some algebraic manipulation and possibly a trigonometric substitution. Once integrated, you'll get the arc length ( L ).

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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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- What is the surface area produced by rotating #f(x)=1/(x^2+1), x in [0,3]# around the x-axis?
- What is the arclength between two points on a curve? How does it differ from the distance?

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