# Solve intersection

• Apr 28th 2009, 11:32 AM
Grep
Solve intersection
This is a calculus problem, but the part with which I'm concerned is probably more appropriate in this forum.

Ok, I need to find the area bounded by these two equations:

$y = \cos {x}$
and
$y = 1 - \frac {2x}{\pi}$

First, I need to find the points at which they intersect so I can find the limits of integration. So making them equal to each other, and rearranging:

$\cos x = 1 - \frac {2x}{\pi} \implies \cos x - 1 + \frac {2x}{\pi} = 0$

Now, by looking at the graph, I can tell that the two intersect at $(0, 1), (\frac {\pi}{2}, 0)$ and $(\pi, -1)$. However, can this be solved explicitly? Or do I need to use something like Newton's method to find these values?

Thanks!
• Apr 28th 2009, 07:21 PM
HallsofIvy
Generally speaking, equations which have the unknown variable both in a transcendental function (exponential, logarithm, trig function) cannot be solved in terms of elementary functions and that is definitely the case here. You will need to use some kind of numerical solution, such as Newton's method.
• Apr 28th 2009, 07:37 PM
Grep
Thanks for the general rule, I was thinking that might often be the case. Certainly, it resisted my best efforts, and Sage (my CAS) found no better solution than I.

Newton's method it is, then. And thanks again for your response!