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Definitions from Fubini and iterated integrals

All definitions from Fubini and iterated integrals.

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[DEFINITION]

Cross-section integral

For a function $f$ on $R=[a,b]\times[c,d]$ , when the one-variable integrals exist, the vertical and horizontal cross-section integrals are

C_{\mathrm{vert}}(x)=\int_c^d f(x,y)\,dy, \qquad C_{\mathrm{horiz}}(y)=\int_a^b f(x,y)\,dx.

They measure the signed areas of slices obtained by fixing $x$ or $y$ .

When $f\ge0$ , these are ordinary cross-sectional areas. When $f$ takes both signs, negative parts are subtracted.

[1]:Fubini's theorem on a rectangle

This page builds on Double integrals and Riemann sums.

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Cross-section integral