Website title: Formal Relation Between Integrals of y=f(x) and y=x

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SUMMARY

The discussion establishes a formal relationship between the integrals \(\int yx \, dx\) and \(\int x y \, dy\) when \(y = f(x)\) over the interval \(x \in [a, b]\). It confirms that for the indefinite integral, the equation \(\int x y \, dy = \int x y y' \, dx\) holds true. Additionally, it emphasizes the importance of correctly interpreting limits in definite integrals, as the limits for the \(dx\) integral pertain to \(x\) while those for the \(dy\) integral pertain to \(y\).

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JulieK
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Is there a formal relation that links
\int yxdx OR \int_{a}^{b}yxdx
with
\int xydy OR \int_{a}^{b}xydy
where y=f(x) over the interval x\in\left[a,b\right].
 
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JulieK said:
Is there a formal relation that links
\int yxdx OR \int_{a}^{b}yxdx
with
\int xydy OR \int_{a}^{b}xydy
where y=f(x) over the interval x\in\left[a,b\right].

For the indefinite integral (assuming everything is nice):
\int xydy =\int xyy'dx

For the definite integral there is a a similar relationship, but be careful with the limits. For the dx integral, the limits refer to x, while for the dy integral, the limits refer to y.
 

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