# Calculus

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.

1. 👍 0
2. 👎 0
3. 👁 1,582
1. 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

1. 👍 1
2. 👎 5

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