Area under the curve visually We know the area function $A(x)$ is an antiderivative of $f(x)$.Meaning there could be numerous area functions whose derivative will be $f(x)$.Could you please show 2 area functions in graph which only differ by a constant?
 A: Consider the function $f(x)=2x$. This has an area function describing the signed area under the curve from $0$ to $x$ given by: $A_0(x)=\int_0^x2t\,\mathrm dt=x^2$. It also has another area function describing the signed area under the curve from $1$ to $x$ given by $A_1(x)=\int_1^x2t\,\mathrm dt=x^2-1$. The constant difference of $1$ corresponds to the area of the triangle with vertices $(0,0)$, $(1,0)$ and $(1,2*1)=(1,2)$. Both of these area functions are two different antiderivatives of $f(x)=2x$, and all (real-valued) antiderivatives arise in this way.
The following graph may help show what's going on here:

A: By integrating $ y= \cos x $ we obtain $ y= \sin x + C$ where $C$ is an arbitrary constant in $y$
The extra area added is the arbitrary constant multiplied by their domain interval.
$$ \int y(x) dx + C (x_2-x_1) $$
When we say area under a curve, it is implied that the constant of integration is zero.
Graph shows difference between two integrands yellow patch which arise from constants $ C_1=0, C_2=2 $ for $ y,y_2$
$$ y_1= \cos x  + 1, y_2= \cos x +2$$
The wavy region can be made to descend as shown to an equal area rectangular patch as shown.. where each ordinate goes down to touch the x- axis.

