Problem 58 - Calculus of a Single Variable

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 \((\overline{x}, \overline{y})\) of \(V .\)

Step-by-Step Solution

Answer

The centroid \((\overline{x}, \overline{y})\) of \(V\) is \( (0, \frac{8}{3}) \)

Step 1: Define the region

First, understand the region \(V\). The condition \(y \leq 4\) defines a horizontal line above which the region cannot extend. The condition \(|x| \leq y \leq |x|+3\) tells that we have two triangles, one where \(x \geq 0\) and other where \(x < 0\), both with base \(-3 \leq y \leq 3\). Hence, we have two identical isosceles triangles, just mirrored over the y-axis.

Step 2: Calculate the area

The next step is to find the area of this region. An isosceles triangle's area can be calculated as \(A = \frac{1}{2}*base*height\). Since there are two triangles, the total area is \(2*\frac{1}{2}*3*4 = 12\).

Step 3: Calculate the x-coordinate of the centroid

The x-coordinate of the centroids of the two triangles are 0 (as they are symmetrical over the y-axis). Hence, the x-coordinate of the centroid of the region \(V\) is also 0.

Step 4: Calculate the y-coordinate of the centroid

The y-coordinate of a triangle's centroid is \(\frac{2}{3}\) times its height, in our case it will be \(\frac{2}{3}*4 = \frac{8}{3}\). Since both triangles are identical, the average centroid y-coordinate would remain the same for the region \(V\), which is \(\frac{8}{3}\).