Solution to Problem 12.3-3© 2005 by Karl Hahn |
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The problem is to find the area bounded by the y-axis, the curve, y = cos(x), and the line, y = x. Again one of the integration limits is already given by having the area bounded by the y-axis. So we have only one intersection point to find. But as stated in the main text, you will have to use Newton-Raphson iteration to approximate the solution to cos(x) - x = 0.
We use the xn+1 = xn - f(xn)/f'(xn) formula where f(x) = cos(x) - x and f'(x) = -sin(x) - 1. Hence
xn+1 = xn + |
cos(xn) - xn |
So what to pick for the first guess, x0? Even without looking at the graph it's pretty clear that the intersection point is between x = 0 and x = 1. This is because cos(0) = 1, cosine is decreasing in the first quadrant, and cos(x) ≤ 1 in the first quadrant. This means that the line, y = x, is already greater than y = cos(x) by the time it reaches the point, (1,1). So here I have chosen x0 = 0.5 (note that you could have chosen any 0 < x0 < π/2 and the iteration would still rapidly converge). Iterating the formula you get
x0 = 0.500000000000000 x1 = 0.755222417105636 x2 = 0.739141666149879 x3 = 0.739085133920807 x4 = 0.739085133215161 x5 = 0.739085133215161The integral, then, is
A = |
0.739085133215161
|
cos(x) - x dx = |
sin(x) - |
x2 |
0.739085133215161 |
0
0
A = (0.400488612113379 - 0) ± ξ, 0 ≤ ξ ≤ 5×10-16 |