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.

Question

anonymous · Answer

Draw a graph first

anonymous · Answer

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I know it looks like this

anonymous · Answer

But i really don't know where to go from here.

danjs · Answer

Find the area of the bound region from -1 to 0, and the integrate from -1 to 'a' and set that equal to half the area

danjs · Answer

x=a will be the vertical line that divides it into 2 equal areas

anonymous · Answer

Do you know how to find the area between the curve and the line?

anonymous · Answer

The top will be $x=a$ and the bottom will be $x=y^2-1$.

anonymous · Answer

The bounds of integration will be when the lines intersect

anonymous · Answer

Yea but don't we have split this one in two because if you just take the definite intergral you would get zero

anonymous · Answer

$$
a = y^2-1 \implies 0=y^2-(1+a) \implies 0=(y-\sqrt{1+a})(y+\sqrt{1+a})
$$This is different of squares.

freckles · Answer

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