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.
Best Answer
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 $a = 2 \sqrt[3]{2} \cdot 3^{2/3}-1$.