# Find the value of constant a, given area between parabola and x-axis.

Given $$f\left(x\right)=-x^2+2x+a$$ find the exact value of constant $$a$$ when the area between the curve and x-axis is 16.

I've tried several methods and all of them seem to be too complicated (at least for me) so possibly there is something that I'm missing. What I have tried is finding the x-interception points, which turned out to be $$\sqrt{a+1}+1$$ and $$-\sqrt{a+1}+1$$.

Given that $$x=1$$ divides the parabola into two equal sized areas we would get $$\int _1^{\sqrt{a+1}+1}\left(-x^2+2x+a\right)\:dx=8$$ which would be too hard for me to solve without straight up using a calculator. I also tried forming a system of linear equations, but it also turned out to be complicated as well.

• Try the substitution $u = x -1$ Commented Dec 18, 2018 at 20:47

The limits are indeed at $$x = 1 \pm \sqrt{a+1}$$. So solve:
$$\int\limits_{x = 1 - \sqrt{a +1}}^{1 + \sqrt{a+1}} -x^2 + 2 x + a\ dx = 16$$
and find $$a = 2 \sqrt[3]{2} \cdot 3^{2/3}-1$$.