Finding a constant in a function, given its definite integral. 
If the surface area bounded by the function $f(x)=3x^{2}+4x+k, y=0,x=2$ and $x=3$  is equal to $25$, find $k$.

I asked this question before, but I didn't explain it sufficiently, so sorry for the repetition.
The problem with this question is to determine whether the function has any roots in the period $[2,3]$. Because if I supposed that there are no zeros in this period, I can solve this equation for $k$.
$$\int_{2}^{3}(3x^{2}+4x+k) dx=25$$
and 
$$\int_{2}^{3}(3x^{2}+4x+k) dx=-25$$
but what if the function has its roots in this period, then the integral that I must evaluate would be like this:
$$\int_{2}^{first-root}(3x^{2}+4x+k)dx+\int_{second-root}^{3}(3x^{2}+4x+k)dx=25$$
any ideas?
 A: $f(x)=3x^2 + 4x + k$ 
For any $k, f(x)$ is increasing over the interval.
There could be a maximum of one root inside the interval $(2,3)$
If there is a root $r$ inside the interval, then the area is $|\int_2^r f(x)\ dx| + |\int_r^3 f(x)\ dx|$ 
This area will be less than $f(3) - f(2)$  (or the maximal distance above 0 + the maximal distance below 0 times the size of the inteval.
$f(3)-f(2) = 19$
There is no $k$ such that there is a region of area $25$
Update
Suppose we choose $k$ such that $f(x)$ has a root in the interval $(2,3)$
Our area is the green region below: 

But the area of the green region is less than the area of the two colored regions combined.
And that area is 19.
A: Note that
$$
\int_2^3 \left(3x^2+4x+k\right)dx = 29 + k,
$$
so if there are no roots, we will have to have $k=-4$, and the function becomes $f_+(x) = 3x^2+4x-4$, which is at least $15$ anywhere on $[2,3]$. Hence $k=-4$ is the correct result.
Here is the plot of $f_+(x)$.


Another alternative is to take the entire curve down below the $y$-axis, in which case you want $29+k=-25 \iff k = -54$, in which case you can see the maximum value of $f_-(x) = 3x^2+4x-54$ on $[2,3]$ is below $-15$, again having no roots. So $k=-54$ is another correct result in this interpretation.
Here is the plot of $f_-(x)$.

