Let P cosh ob sinh b be the point on the hyperbola orrespond

Let P (cosh ob, sinh b) be the point on the hyperbola orresponding to r(b) for ob> 0. Use the formula for area dy y dx to verify that the area of the region shown in the figure is job.

Solution

x = cosh, y = sinh
dx = sinh, dy = cosh
xdy = cosh^2d
ydx = sinh^2d
putting values in A
A = 1/2 integral of [cosh^2 - sinh^2]d
A = 1/2 integral of d [because cosh^2 - sinh^2 = 1]
A = 1/2 ......hence proved