a-maths~!~!~

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Oct08/Crazyint1.jpg

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The three equations provided are y=8x, y=x^2 and x^2y=1
It is easy to find that A(1/2,4), B(1,1), C(8,64)
The area of OCO is
8*64/2-∫x^2 dx [fom 0 to 8]
=256-x^3/3 [from 0 to 8]
=256-512/3
To find the area of OAB, draw a horizontal line from B, then
consider y=8x, x=y/8 ; y=x^2,x=√y ; x^2y=1, x=y^(-1/2)
So the area of OAB is
∫√y dy [from 0 to 1] - (1/8)/2 + ∫y ^(-1/2) dy [from 1 to 4] - 3/2
=2/3-1/16+2-3/2
=2/3-1/16+1/2
=(32-3+24)/48
=53/48
So the area of shaded region
=256-512/3-53/48
=4043/48