Problem 58 - Essential Calculus

Question

Find the centroid of the region determined by the graphs of the inequalities. $$ y \leq \frac{1}{4} x^{2},(x-4)^{2}+y^{2} \leq 16, y \geq 0 $$

Step-by-Step Solution

Answer

After analyzing the problem, identifying the intersection region of the inequalities, expressing this area in polar coordinates, and proceeding with the double integrals, the centroid will be computed.

Step 1: Understand the Inequalities

We have three inequalities: \(y \leq \frac{1}{4} x^{2}\) which is a parabola opening upwards; \((x-4)^{2} +y^{2} \leq 16\), which is a circle with radius 4 and center at (4, 0); and \(y \geq 0\), which is the region above the x-axis. We will look for the region that satisfies all three inequalities.

Step 2: Identify the Region

The intersection of these requirements is a lens-shaped region. On the X-axis, it extends from x=-4 to x=4, and on the Y-axis, from y=0 up to the parabola. The upper boundary is the parabola, the bottom boundary is the x-axis, and the side boundaries are the circle. We need to double integrate over this region to find the centroid.

Step 3: Calculate the Necessary Integrals

We will calculate:1. The double integral of 1 over the region, to find the area.2. The double integral of x over the region, to find the x-coordinate of the centroid.3. The double integral of y over the region, to find the y-coordinate of the centroid.Given the symmetry of the problem, it makes sense to use polar coordinates for the integration. To do this, note that the polar coordinates of the points in the region trace out the angles from \(\theta = -\pi/2\) to \(\theta = \pi/2\) and the radius from \(r = 0\) to \(r = 2\cos(\theta)\)

Step 4: Calculate the Centroid

To calculate the centroid, divide the second integral by the first to get the x-coordinate, and the third integral by the first to get the y-coordinate. The centroid \((\bar{x}, \bar{y})\) is given by: \(\bar{x} = \frac{\int \int_R x dA}{\int \int_R dA}\), \(\bar{y} = \frac{\int \int_R y dA}{\int \int_R dA}\) where R is the region of interest.After replacing \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) in the integrals and solving, we calculate the centroid coordinates.