Question

Use Green's Theorem to verify the line integral formulas. The centroid of the region having area \(A\) bounded by the simple closed path \(C\) is \(\bar{x}=\frac{1}{2 A} \int_{C} x^{2} d y, \quad \bar{y}=-\frac{1}{2 A} \int_{C} y^{2} d x\).

Short answer

The coordinates of the centroid using Green's theorem to compute the line integrals, resulting to \( \bar{x}=\frac{1}{A} \int_{D} x d (x, y) \) and \( \bar{y}=\frac{1}{A} \int_{D} y d (x, y) \).

Step-by-step solution

Understanding Centroid Formulae and Line Integral

The centroid of a region is the center of mass or the 'average' of the region. The formulas given for \( \bar{x} \) and \( \bar{y} \) are based on line integrals which needs to be computed along a curve \( C \). These integral formulas will provide the x and y coordinates of the centroid.

Using Green's Theorem

Next step is to compute the line integrals using Green's Theorem. The Green's Theorem relates a line integral around a simple closed curve \( C \) to a double integral over the plane region \( D \) bounded by \( C \). Applying Green's theorem to \( \int_{C} x^{2} d y \) and \( \int_{C} y^{2} d x \), We derive: \( \int_{C} x^{2} d y = \int_{D} div(x^{2}, 0) d A = 2\int_{D} x dA \) and \( \int_{C} y^{2} d x = \int_{D} div(0, -y^{2}) d A = -2\int_{D} y dA \).

Compute Centroid Formulas

Substituting the derived integrals in the formulas for the centroid, we obtain : \( \bar{x}=\frac{1}{A} \int_{D} x d A = \frac{1}{A} \int_{D} x d (x, y) \) and \( \bar{y}=\frac{1}{A} \int_{D} y d A = \frac{1}{A} \int_{D} y d (x, y) \), these are the coordinates of the centroid using Green's theorem to compute the line integrals.