Hello, zer0e!
It should be simple Algebra . . .(okay, maybe not so simple).
Find the area of the plane region bounded by the curve
and the tangent line to the curve at the point (1,1). Your tangent line is correct: .
To find the intersections, set the functions equal to each other:
. .
We see that is a solution, so is a factor.
With long division: .
The intersections occurs at: . Code:

 ...♥(1,1)
 ...o::*
     ..o:::::*    
o:::::::*
.:::::::*
o:::::::*
.::::::* 
o::::* 
:::* 
:* 
♥(2,8) 

Therefore: .