Find the number b such that the line y = b divides the region bounded by the curves x = y2 − 1 and the y-axis into 2 regions with equal area. Give your answer correct to 3 decimal places.

Question

aum · Answer

$x=y^2-1$ is symmetric about the x-axis because replacing y by -y yields the same equation.  So I think the x-axis, or the equation y = 0, divides the regions into two equal areas.  So b = 0, unless I am misunderstanding the question.

anonymous · Answer

Thanks for the explanation, I just graphed it for myself and it seems that b=0 to me as well

aum · Answer

But if the region they are talking about is bounded by x = y^2 - 1, the y-axis and above the x-axis, then it is a different problem.

anonymous · Answer

b=0 was incorrect :/ I think it's only the region above the x-axis

aum · Answer

Then you have to set up two integrals to find the two areas, equate them and solve for b.

aum · Answer

$$
\int_b^1xdy = \int_0^bxdy \
\int_b^1(y^2-1)dy = \int_0^b(y^2-1)dy \
$$

aum · Answer

It is a cubic equation and will have three solutions.  Choose the one where 0 < b < 1.
Give answer to 3 decimal places.

anonymous · Answer

Thank you!!!!!

aum · Answer

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