Question

Let \(V\) be the region in the cartesian plane consisting of all points \((x, y)\) satisfying the simultaneous conditions \(|x| \leq y \leq|x|+3\) and \(y \leq 4\) Find the centroid \((\bar{x}, \bar{y})\) of \(V\).

Short answer

The centroid \((\bar{x}, \bar{y})\) of region V is \((0, 25/14)\)

Step-by-step solution

Identify the Region

The region \(V\) is made up of a triangle and a square. The square is defined by \(-y \leq x \leq y\), and \(0 \leq y \leq 3\), while the triangle is defined by \(-y \leq x \leq y\), and \(3 \leq y \leq 4\).

Compute Area of the Region

For the square region, we have that its area \(A_{sq}\) is obtained by integrating \(2y\) (as \(y\) goes from 0 to 3) which is \(9\). For the triangle region, we have that its area \(A_{tr}\), is derived by integrating \(2y - 2y^2/2\) (as \(y\) goes from 3 to 4) which gives \(5\). Adding these two gives us the total Area \(A = 9 + 5 = 14\).

Compute Centroid Coordinates

Now we compute the x and y coordinates of the centroid using the formulas for \(\bar{x}\) and \(\bar{y}\). To compute \(\bar{x}\), we find that since the region is symmetric about y-axis, the x-coordinate of the centroid \(\bar{x}\) will be 0. To compute \(\bar{y}\), we integrate \(y \cdot 2y\) for the square (as \(y\) goes from 0 to 3) which gives 18, and \(y[-2y + y^2]\) for the triangle (as \(y\) goes from 3 to 4) which gives 7, and summing up these quantities divided by the total area 14 gives \(\bar{y} = 25/14\).