Can Abel Integral Equations Determine a Curve from Known Area Under It?

[ber70]

If we know area under the curve, are we able to find the curve using Abel integral equations?

 

[quantumdude]

Clearly not. Let a region be bounded by the lines $x=0$, $x=1$, $y=0$, and the graph of $y=f(x)$. Let $f(x)\geq 0$ on $\[0,1\]$, and let the area of the region be 1. There is no unique solution for $f(x)$, as each of the following work:

$$f_1(x)=1$$

$$f_2(x)=2x$$

$$f_3(x)=\frac{\pi}{2}\sin(\pi x)$$

 

[ber70]

Does first type Volterra equation work?

 

[quantumdude]

Did you comprehend my answer to your first question at all?

 

[ber70]

Question:"Find the curve (or the function) y(x) for which the area under the curve is equal
to $$\frac{1}{\pi }$$th of the area formed by the rectangle whose one side is x and the other side is y(x)."

I think $$\int _a^xy(t)dt=\frac{1}{\pi }y(x).x$$

give me the curve.

 

[quantumdude]

to $$\frac{1}{\pi }$$th of the area formed by the rectangle whose one side is x and the other side is y(x)."

You're dealing with a rectangle that has only two sides, one of which is not necessarily a straight line segment? Seriously?