Calculus Area question?

Find the point of tangency (a,b) from the observer's line of sight to the spherical cap - note, the observer is at (0,r+H):

http://img98.imageshack.us/img98/4983/circlelr5.jp...

m = (b - r - H) / a ........ slope of the line of sight

b = √(r² - a²) ............... from the equation of the circle x² + y² = r²

m = -a / √(r² - a²) ........ slope of the circle at the point of tangency

(r + H - √(r² - a²)) / a = a / √(r² - a²) ....... solve for a with m = m

√(r² - a²) = r² / (r + H)

a = r√(H² + 2rH) / (r + H)

b = √(r² - a²)

= √(r²(r + H)² - r²(H² + 2rH)) / (r + H)

= r² / (r + H)

Area of a spherical cap = 2πrh ......... note: h = r - b

= 2πr (r - r² / (r + H))

= 2πr²H / (r + H)

proved.

Answer: see above