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Compute the area of the region bounded by:

$\displaystyle y=\cos(x)$

$\displaystyle y = x$

$\displaystyle x = 0$

I puzzled for a bit, did some calculations, but could not get away from using a numeric root-finding method for:

$\displaystyle f(x)=\cos(x)-x=0$

to determine the upper limit of integration.

I was curious if someone here might have an insight I missed. By the way, both people that responded also approximated the root.

On a side note, I recall seeing once that a simple method for approximating this root is as follows:

Make sure your calculator is in radian mode.

Enter any number on your calculator.

Take the cosine of this result.

Keep successively taking the cosine of the results, and your calculator will converge (slowly) to the desired root.

I can see why this works. If

*r*is the root, then we have both:

$\displaystyle r=\cos(r)$

$\displaystyle r=\cos^{-1}(r)$

And from this we have:

$\displaystyle r=\cos(\cos(\cos\cdots\cos(r)))$