Çözüldü Absolute Value Function - Quadratic Curves - Derivative - Integral - Trapezoid Area

For the science high school classic exam:

Find the answer to the following question in two different solutions:
What is the area between y = f(x) = x·|x - 3| + 4 and its tangent at x = 2 for x equal to or greater than 0 and less than or equal to 2?

x < 3 ⇒ y = f(x) = x( -(x - 3) ) + 4 = -x^2 + 3x + 4
f '(x) = -2x + 3 ⇒ f '(2) = -2·2 + 3 = -1
f(2) = -2^2 + 3·2 + 4 = 6
Equation of the Tangent Line: y - 6 = -1·(x - 2) ⇒ y = h(x) = -x + 8
Note: x ≥ 3 ⇒ y = f(x) = x(x - 3) + 4 = x^2 - 3x + 4

Solution - 1:

https://i72.servimg.com/u/f72/19/97/10/39/absolu10.png
https://www.wolframalpha.com/input?i=integrate (-x+8-(x*abs(x-3)+4)) dx from x=0..2

Solution - 2:

https://i72.servimg.com/u/f72/19/97/10/39/absolu11.png
https://www.wolframalpha.com/input?i=((8+6)/2)*2-(integrate (-x^2+3x+4) dx from x = 0..2)

Graph:

https://i72.servimg.com/u/f72/19/97/10/39/mutlak29.png

Source (Turkish): https://matematiknotu.com/wp-content/uploads/2021/04/Turevin-Geometrik-Yorumu-Testi-1.pdf
(The last question, Answers on the upper right side of the page)

Note: Only the slope of the tangent is asked, so it is unnecessary to find the value of "k" which is the ordinate of the tangency point.