# 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?

• What do you call an area function, precisely ?
– user65203
Feb 23, 2021 at 13:13
• I mean suppose we define $A(x)=\int_{a}^{x}f(t)dt$,then $A(x)$ denotes the area from $a$ to $x$.We know $A'(x)=f(x)$. Feb 23, 2021 at 13:43
• Then the area function is unique.
– user65203
Feb 23, 2021 at 13:44
• @YvesDaoust It's not unique, since there's the parameter $a$. If $A_a(x)=\int_a^xf(t)\,\mathrm dt$, then $A_0$ and $A_1$ would be different functions. Feb 23, 2021 at 13:46
• Why is it fixed?I mean since A'(x)=f(x),aren't there infinitely many anti derivatives? Feb 23, 2021 at 14:04

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:

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.

• Could you please tell me which is the extra added area? Feb 23, 2021 at 13:36
• Added yellow patch between two arbitrary constants Feb 23, 2021 at 13:40
• The yellow patch is not area under the cosine graph, and so does not seem relevant. Feb 23, 2021 at 13:45
• I was trying to illustrate area between two arbitrary constants $(1,2)$. Hope now it is better. Feb 23, 2021 at 14:01
• That is a region between the two functions which differ by a constant, and sometimes "area" is used in English to mean "region", but the question is about integrals that measure the (signed) value of area under a curve. In your example, it would be (signed) areas under the cosine curve that are relevant. Feb 23, 2021 at 14:19