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?
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2$\begingroup$ What do you call an area function, precisely ? $\endgroup$– user65203Feb 23, 2021 at 13:13
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1$\begingroup$ 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 $A'(x)=f(x)$. $\endgroup$– a_i_rFeb 23, 2021 at 13:43
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$\begingroup$ Then the area function is unique. $\endgroup$– user65203Feb 23, 2021 at 13:44
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$\begingroup$ @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. $\endgroup$– Mark S.Feb 23, 2021 at 13:46
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1$\begingroup$ Why is it fixed?I mean since A'(x)=f(x),aren't there infinitely many anti derivatives? $\endgroup$– a_i_rFeb 23, 2021 at 14:04
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2 Answers
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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:
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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.
answered Feb 23, 2021 at 13:33
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$\begingroup$ Could you please tell me which is the extra added area? $\endgroup$– a_i_rFeb 23, 2021 at 13:36
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$\begingroup$ Added yellow patch between two arbitrary constants $\endgroup$ Feb 23, 2021 at 13:40
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$\begingroup$ The yellow patch is not area under the cosine graph, and so does not seem relevant. $\endgroup$– Mark S.Feb 23, 2021 at 13:45
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$\begingroup$ I was trying to illustrate area between two arbitrary constants $(1,2)$. Hope now it is better. $\endgroup$ Feb 23, 2021 at 14:01
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$\begingroup$ 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. $\endgroup$– Mark S.Feb 23, 2021 at 14:19