Calculating energy absorbed by an element
Let's start with your second question.


  2- What does area under the graph of the power represent?


The axes are power and time so the area under the graph is power x time = energy and the units will be Ws (watt-seconds).

Now back to the first question:


  1- How the mathematical equation is derived and why the limits are chosen in the way they are give in the solution?


The relationship between power, voltage and current is given by P = VI. We can use this to draw the curve C.


At t = 0: V = 0 so P = 0.
At t = 3: I = 2, V = 4, so P = VI = 4 x 2 = 8 W.
At t > 3: I = 1, V = 4, so P = VI = 4 x 1 = 4 W.


Looking at the triangular part:




The area under the curve at time t is given by the area of the triangle. The base is t and the height mt, where m is the slope of the line which can be shown to be 8/3.
Therefore, the area under the curve at t is given by \$ \frac {1}{2}bh = \frac {1}{2}t \times \frac {8}{3}t = \frac {4}{3}t^2 \$.


That's the more difficult part. You should be able to work out the second part.