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Theory and Modern Applications

Figure 5 | Advances in Difference Equations

Figure 5

From: The curl theorem of a triangular integral

Figure 5

Two kinds of integrals for a monotonically increasing function.

An integral along x-axis is ∫ y = f ( x ) [ A , B ] y d x = lim n → ∞ ∑ k = 1 n y k Δ x k

and that along y-axis is ∫ x = f - 1 ( y ) [ A , B ] x d y = lim n → ∞ ∑ k = 1 n x k Δ y k .

It holds ∫ y = f ( x ) [ A , B ] y d x + ∫ x = f - 1 ( y ) [ A , B ] x d y = x B y B - x A y A ,

where yA = f(xA) and yB = f(xB).

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