Find the area of the region bounded by the parabola $y = 2x^2$, the tangent line to this parabola at $(3, 18)$, and the $x$-axis. Is my answer correct? 
$f(x) = 2x^2 \gets$ this is the parabola 
$f(3) = 2 \times 9 = 18 \to$ the parabola passes through $A (3 ; 18$), so its tangent line does too.
$f'(x) = 4x \gets$ this is the derivative 
…and the derivative is the slope of the tangent line to the curve at $x$
$f'(3) = 4 \times 3 = 12 \gets$ this is the slope of the tangent line to the curve at $x = 3$
Equation of the tangent line 
The typical equation of a line is: $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.
We know that the slope of the tangent line is $12$. 
The equation of the tangent line becomes $y = 12x + b$.
The tangent line passes through $A (3 ; 18)$, so these coordinates must satisfy the equation of the tangent line. 
$y = 12x + b$
$b = y - 12x \to$ I substitute $x$ and $y$ by the coordinates of the point $A (3 ; 18)$.
$b = 18 - 36 = - 18$
$\to$ The equation of the tangent line is $y = 12x - 18$.
Intersection between the tangent line to the curve and the $x$-axis: $\to$ when $y = 0$.
\begin{align}
y &= 12x - 18 \to \text{when } y = 0 \\
12x - 18 &= 0 \\
12x &= 18 \\
x &= 3/2
\end{align}
$\to$ Point $B (3/2 ; 0)$
Intersection between the vertical line passes through the point $A$ and the $x$-axis: $\to$ when $x = 3$.
$\to$ Point $C (3 ; 0)$
The equation of the vertical line is $x = 3$.
Area of the region bounded by the parabola $y = 2x^2$, the tangent line to this parabola at $(3; 18)$, and the $x$-axis. 
$=$ (area of the region bounded by the parabola $y = 2x^2$ and the $x$-axis) minus (area of the triangle $ABC$) 
$=$ [integral (from $0$ to $3$) of the parabola] minus $[(x_C-x_B)\cdot(y_A-y_C)/2]$ 
\begin{align}
&= \int_0^3 2x^2 dx -\frac{(x_C-x_B)(y_A-y_C}{2}  \\
&= \left. \frac23 x^3 \right|_0^3 -\frac{(3-3/2)(18-0)}{2}  \\
&= \frac23 \cdot 3^2 -(6/2 - 3/2)\cdot9  \\
&= \frac23 \cdot 27 -\frac32 \cdot 9  \\
&= 18 - \frac{27}{2}  \\
&= \frac{36}{2} -\frac{27}{2}  \\
&= \frac92 \text{ square units}
\end{align}

$y=2x^2$ has derivative $y'=4x$. So the derivative will be positive for all $x>0$. This means the tangent line will intersect the x-axis at some point $x_a<x$ for a given x. 
This means the area under the parabola and bounded by the tangent line will have two separate regions, A region with $x<x_a$ which is made up of only area under the parabola, and a region $x>x_a$ where area is from the area between the parabola and the tangent to $(3,18)$.
The equation of a tangent line at $(x_0,y_0)$ is $$\frac{y-f(x_0)}{x-x_0}=f'(x_0)$$
Or :  $y=f'(x_0)(x-x_0)+f(x_0)$
The x intercept happens where $y=0$.
Requiring $y=0$ implies an x intercept of $x_c=\frac{-f(x_0)}{f'(x_0)}+x_0$
So from the above arguments with $x_0=3$:
$$A=\int_0^{x_c}2x^2dx+\int_{x_c}^32x^2-(12x-18)dx$$
But this can be simplified. The area under the tangent line is a triangle. So the the parabola can be integrated ignoring the tangent line, and then subtracting the area of the triangle. 
From the above, we know $(x_0-x_c)=\frac{f(x_0)}{f'(x_0)}$, the base of the triangle. The height is just $f(x_0)$.
So the area of the triangle is $A_t=\frac{1}{2}\frac{f(x_0)^2}{f'(x_0)}$
So: 
$$A=\int_0^{x_0}f(x)dx-\frac{1}{2}\frac{f(x_0)^2}{f'(x_0)}$$
and solve for $x_0=3$. 
In this form, an expression can be found for $x_0$ which extremizes the area. 
