How do you find the arc length of the curve #y=(5sqrt7)/3x^(3/2)-9# over the interval [0,5]?

Answer 1

#s=int_0^5sqrt(1+175/4x)color(white).dx=1/525(879^(3/2)-8)#

The arc length of a function #y# on the interval #[a,b]# is given by:
#s=int_a^bsqrt(1+(dy/dx)^2)color(white).dx#

Here:

#y=(5sqrt7)/3x^(3/2)-9#
#dy/dx=(5sqrt7)/3(3/2x^(1/2))=(5sqrt7)/2x^(1/2)#

Then the arc length desired is:

#s=int_0^5sqrt(1+((5sqrt7)/2x^(1/2))^2)color(white).dx#
#s=int_0^5sqrt(1+175/4x)color(white).dx#
Let #u=1+175/4x#, which implies that #du=175/4dx#. Moreover, note the change of bounds of the integral under this substitution: #x=0=>u=1# and #x=5=>u=879/4#. The integral is then:
#s=4/175int_1^(879//4)u^(1/2)color(white).du#

Integrating using the power rule for integration:

#s=4/175(2/3u^(3/2))|_1^(879//4)#
#s=8/525((879/4)^(3/2)-1^(3/2))#
#s=8/525(879^(3/2)/8-1)#
#s=1/525(879^(3/2)-8)#
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Answer 2

To find the arc length of the curve ( y = \frac{5\sqrt{7}}{3}x^{\frac{3}{2}} - 9 ) over the interval ([0,5]), follow these steps:

  1. Compute the derivative of ( y ) with respect to ( x ).
  2. Use the formula for arc length:

[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx ]

  1. Plug in the derivative into the formula.
  2. Evaluate the integral over the given interval ([0,5]).
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Answer 3

To find the arc length of the curve y = (5√7)/3x^(3/2) - 9 over the interval [0, 5], we use the arc length formula:

Arc Length = ∫√[1 + (dy/dx)^2] dx

First, we find dy/dx by taking the derivative of y with respect to x.

Then, we plug this derivative into the formula and integrate over the given interval [0, 5].

Finally, we evaluate the integral to find the arc length.

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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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