Editing Centroid (section)

=== By integral formula ===
The centroid of a subset <math>X</math> of <math>\R^n</math> can also be computed by the vector formula

<math display=block>C = \frac{\int x g(x) \ dx}{\int g(x) \ dx} = \frac{\int_X x \ dx}{\int_X dx}</math>

where the [[integral]]s are taken over the whole space <math>\R^n,</math> and <math>g</math> is the [[Indicator function|characteristic function]] of the subset <math>X</math> of <math>\R^n \! : \ g(x) = 1</math> if <math>x \in X</math> and <math>g(x) = 0</math> otherwise.<ref name = protter526>{{harvtxt|Protter|Morrey|1970|p=526}}</ref>  Note that the denominator is simply the [[Measure (mathematics)|measure]] of the set <math>X.</math> This formula cannot be applied if the set <math>X</math> has zero measure, or if either integral diverges.

Alternatively, the coordinate-wise formula for the centroid is defined as

<math display=block>C_k = \frac{\int z S_k(z) \ dz}{\int g(x) \ dx},</math>

where <math>C_k</math> is the <math>k</math>th coordinate of <math>C,</math> and <math>S_k(z)</math> is the measure of the intersection of <math>X</math> with the [[hyperplane]] defined by the equation <math>x_k = z.</math>  Again, the denominator is simply the measure of <math>X.</math>

For a plane figure, in particular, the barycentric coordinates are

<math display=block>
C_{\mathrm x} = \frac{\int x S_{\mathrm y}(x) \ dx}{A}, \quad
C_{\mathrm y} = \frac{\int y S_{\mathrm x}(y) \ dy}{A},
</math>

where <math>A</math> is the area of the figure <math>X,</math> <math>S_{\mathrm y}(x)</math> is the length of the intersection of <math>X</math> with the vertical line at [[abscissa]] <math>x,</math> and <math>S_{\mathrm x}(y)</math> is the length of the intersection of <math>X</math> with the horizontal line at [[ordinate]] <math>y.</math>

==== Of a bounded region ====
The centroid <math>(\bar{x},\;\bar{y})</math> of a region bounded by the graphs of the [[continuous function]]s <math>f</math> and <math>g</math> such that <math>f(x) \geq g(x)</math> on the interval <math>[a, b],</math> <math>a \leq x \leq b</math> is given by<ref name="protter526" /><ref>{{harvtxt|Protter|Morrey|1970|p=527}}</ref>

<math display=block>\begin{align}
\bar{x} &= \frac{1}{A}\int_a^b x\bigl(f(x) - g(x)\bigr)\,dx, \\[5mu]
\bar{y} &= \frac{1}{A}\int_a^b \tfrac12\bigl(f(x) + g(x)\bigr)\bigl(f(x) - g(x)\bigr)\,dx,
\end{align}</math>

where <math>A</math> is the area of the region (given by <math display=inline>\int_a^b \bigl(f(x) - g(x)\bigr) dx</math>).<ref>{{harvtxt|Protter|Morrey|1970|p=528}}</ref><ref>{{harvtxt|Larson|1998|pp=458–460}}</ref>