Find the number a such that the line x = a divides the region bounded by the curves x = y^2 - 1 and the y-axis into 2 regions with equal area. Give your answer correct to 3 decimal places.

Parabola with vertex at x = - 1 and y = 0 then proceeding right above and below the x axis

passing through (0,1) and (0,-1)

∫ left of x = 0 = ∫ right of x = 0

because of symmetry we only need to do + y

y = + / - sqrt (x+1) = + / - (x+1) ^.5

∫ y dx from - 1 to 0

= ∫y dx from 0 to a

∫ y dx = ∫ (x+1) ^.5 dx

= (x+1) ^1.5 / 1.5 at x = - 1 that is 0

at x = 0 that is 1/1.5 = 2/3

so

we need to select upper limit of x = a to get the same area from 0 to a

at x = a

∫ is (a+1) ^1.5/1.5

at x = 0 we know it is 2/3

(a+1) ^1.5 / (3/2) - 2/3 = 2/3

(a+1) ^1.5 / (3/2) = 4/3

(a+1) ^1.5 = 2

1.5 log (a+1) =.301

log (a+1) =.2

a+1 = 1.584

a =.584