Integrating Arcsine: Solving for the Area Between Two Intervals

[tjbateh]

Homework Statement

\int^{1/\sqrt{2}}_{0} \stackrel{arcsinx}{\sqrt{1-x^2}}

Homework Equations

The 0 is supposed to be on the bottom of the intergal, but I could not format it to go there.

The Attempt at a Solution

My attempt was to set u= arcsinx, then DU would equal the bottom, so it would be \stackrel{u}{du}...but from there I get stuck. I know the answer is supposed to be .308...Help anyone?

 

[Gregg]

\int_0^{1\over\sqrt{2}}\frac{\arcsin(x)}{\sqrt{1-x^2}}dx

\int_0^{1\over\sqrt{2}}\frac{\arcsin(x)}{\sqrt{1-x^2}}

If you set u=arcsinx

du = \frac{dx}{\sqrt{1-x^2}}

\int u du = \int {arcsinx\over \sqrt{1-x^2}} dx

 

[tjbateh]

Ok, that makes sense. Then from there do you just sub in the two intervals?

 

[Gregg]

Yeah you need to make sure the intervals are consistent i.e. u=arcsinx, sinu=0 or 1/sqrt(2) quite simple, and that way you won't need to change all back to x.

 

[tjbateh]

If possible, could you set up the substitution for me so I can see how it looks, and then I can work from that?

 

[Gregg]

\int_0^{1\over\sqrt{2}}\frac{\arcsin(x)}{\sqrt{1-x^2}}dx

using the substition:

u = \arcsin(x)

{du\over dx} = {1\over \sqrt{1-x^2}} \Rightarrow du = {dx\over \sqrt{1-x^2}}

\int u du = \int \arcsinx {dx\over \sqrt{1-x^2}}


then make sure the limits are changed:


x_0 = 0, x_1 = {1\over\sqrt{2}}

\arcsin(0) = u_0 = 0

\arcsin({1\over\sqrt{2}}) = u_1 = {\pi\over 4}

\int_0^{1\over \sqrt{2}} {\arcsin(x)\over \sqrt{1-x^2}} dx = \int_0^{\pi\over 4} u du

 

[tjbateh]

ok so


\int^{\Pi/4}_{0} dx/\sqrt{1-x^2}


Then from there I substitute pie/4 in for x?

 

[tjbateh]

or would it be something like this..arcsin (Pie/4) - arcsin (0)??

 

[Gregg]

No. The whole integral has been transformed.

\int_{0}^{\pi\over 4} u du = \int_{0}^{1\over\sqrt{2}} \frac{\arcsin (x)}{\sqrt{1-x^2}} dx

Basically because you swapped x for sin(u) (essentially) if x is 1\over\sqrt{2} then u has to be \pi\over 4 doesn't it? So everything is consistent? 0 stays the same because sin(0)=0.

 

[tjbateh]

True, so we have that set up. Now where do we go from there?

 

[Gregg]

integrate u from 0 to pi/4 and that's it.

 

[tjbateh]

so I would get .903??

The answer in the back of the book for this problem says .308.
Did I do something wrong?

 

[tjbateh]

on calcchat.com, they have something like this for their new integral.

\int^{\frac{1}{\sqrt{2}}}_{0} \frac{1}{2} arcsin²x


then from there they get \frac{\Pi^2}{32}


which approximates to 0.308, but how on Earth did they get that new integral? Does what you see here make any sense?