Q.29 - Calculus

Question

If the density at each point in $\mathcal{T}$ is proportional to the point's distance from the $x$-axis, find the center of mass of $\mathcal{T}$.

Step-by-Step Solution

Answer

Area of the region bounded by the spiral and the $x$-axis is

$A=\frac{{\pi }^3}{6}$

Step 1: Set up the integral

If the density at each point is proportional to the point's distance from an axis, then \(\rho = k \cdot d\) where \(d\) is the distance. The mass is \(M = \iint_T \rho \, dA\).

Step 2: Convert to appropriate coordinates

For a spiral region, polar coordinates are natural: \(d = r\) (distance from origin), \(dA = r\,dr\,d\theta\). Set up and evaluate the double integral over the region bounded by the spiral.