Can anyone explain a method on how to go about this problem? I don't care for the answer; I'm just perplexed on how to begin the problem in a successful manner. Thanks!
edit here's syntax:
Can anyone explain a method on how to go about this problem? I don't care for the answer; I'm just perplexed on how to begin the problem in a successful manner. Thanks!
edit here's syntax:
Where did you get this problem? It is pretty standard Calculus problem- are you not taking Calculus and have not learned this?
If y= f(x), the arclength, from x= a to x= b, is given by $\displaystyle \int_a^b \sqrt{1+ f'^2(x)} dx$.
Thanks! Oh, so you have to find the derivative of y and plug into the length formula you've provided right?
My instructor hasn't covered this (yet?) and gave this as one of a bunch of questions to do for next week. Had no clue what formula to use but thanks for clearing that up!
Original problem:y=sqrt(x-x^2)+arcsin(sqrt(x))
Sorry, this is the derivative of y I got to plug into the Length formula:
sqrt(-(-1+x) x)/x
Does anyone know what the bounds are and how to find them? Do we set the original y equation = to 0?
I think it may be 0 to 1 but have no clue as to why.
$\displaystyle \displaystyle \begin{align*} f(x) &= \sqrt{ x - x^2 } + \arcsin{ \left( \sqrt{x} \right) } \\ \\ f'(x) &= \frac{1 - 2x}{2\,\sqrt{ x - x^2 }} + \frac{1}{2\,\sqrt{x} \, \sqrt{ 1 - x } } \\ &= \frac{ 1 - 2x }{ 2 \, \sqrt{ x \left( 1 - x \right) } } + \frac{1}{2 \, \sqrt{ x \left( 1 - x \right) } } \\ &= \frac{2 - 2x}{2\,\sqrt{x \left( 1 - x \right) }} \\ &= \frac{1 -x}{\sqrt{x \left( 1 - x \right) }} \\ &= \frac{ \sqrt{x \left( 1 - x \right) } }{x} \end{align*}$
So now plugging into the formula for the arclength (I can't see your images so I'm not sure what your bounds should be)...
$\displaystyle \displaystyle \begin{align*} l &= \int_a^b{ \sqrt{ 1 + \left[ f'(x) \right] ^2 } \, dx } \\ &= \int_a^b{ \sqrt{1 + \left[ \frac{\sqrt{x\left( 1 - x \right) }}{x} \right] ^2 } \, dx } \\ &= \int_a^b{ \sqrt{ 1 + \frac{x \left( 1 - x \right) }{x^2} } \,dx } \\ &= \int_a^b{ \sqrt{ 1 + \frac{1 - x}{x} }\,dx } \\ &= \int_a^b{ \sqrt{ \frac{1}{x} } \,dx } \\ &= \int_a^b{ x^{-\frac{1}{2} } \,dx} \\ &= \left[ 2x^{\frac{1}{2}} \right] _a^b \\ &= 2\,\sqrt{b} - 2\,\sqrt{a} \end{align*}$
So now plug in your endpoints...