How do you find the length of the curve #y=sqrt(x-x^2)+arcsin(sqrt(x))#?

Answer 1

#2# units.

The arc length of a continuous curve from #a# to #b# is given by #int_a^b sqrt(1+ (dy/dx)^2)#. Let's start by computing the derivative.
#y' = (1 - 2x)/(2sqrt(x - x^2)) + 1/(2sqrt(x - x^2)#
#y' = (1 - 2x + 1)/(2sqrt(x- x^2))#
#y' = (2 - 2x)/(2sqrt(x - x^2)#
#y' = (2(1 - x))/(2sqrt(x - x^2)#
#y' = (1 - x)/sqrt(x(1 - x))#
Now let's find the endpoints of the function #y#. The function #y = arcsinx# has domain #{x|-1 ≤ x ≤ 1, x in RR}#. However, since the value under the square root has to be positive, #y = arcsinsqrt(x)# has domain #{x| 0 ≤ x ≤ 1, x in RR}#.
The second part of the function, #y = sqrt(x - x^2)#, has the same domain as #y = arcsinsqrt(x)#. So, we can conclude our bounds of integration will be from #0# to #1#. Call the arc length #A#.
#A = int_0^1 sqrt(1 + ((1 - x)/sqrt(x(1 - x)))^2)dx#
#A = int_0^1 sqrt(1 + (1 - x)^2/(x(1 - x)))dx#
#A = int_0^1 sqrt(1 + (1 - x)/x) dx#
#A = int_0^1 sqrt(1 + 1/x - x/x)dx#
#A = int_0^1 sqrt(1 + 1/x - 1)dx#
#A = int_0^1 sqrt(x^-1)#
#A = int_0^1 (x^-1)^(1/2)#
#A = int_0^1 x^(-1/2)#
#A = [2x^(1/2)]_0^1#
#A = 2(1)^(1/2) - 2(0)^(1/2)#
#A = 2#

Hopefully this helps!

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Answer 2

To find the length of the curve given by ( y = \sqrt{x - x^2} + \arcsin(\sqrt{x}) ), you can use the arc length formula for a curve ( y = f(x) ) from ( x = a ) to ( x = b ):

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

First, find the derivative of ( y ) with respect to ( x ) ( ( y' ) ):

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

Now, plug ( y' ) into the arc length formula and integrate from the starting point (( x = a )) to the ending point (( x = b )). Since the curve equation involves square roots and arcsine, the integration may not have a simple closed-form solution and might require numerical methods or special techniques.

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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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