91Ó°ÊÓ

Find the centroid of the region. $$ \text { Top half of the ellipse }\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1 \text { for arbitrary } a, b>0 $$

Find the centroid of the region. $$ \text { Semicircle of radius } r \text { with center at the origin } $$

Compute the surface area of revolution about the \(x\) -axis over the interval. $$ y=\left(4-x^{2 / 3}\right)^{3 / 2}, \quad[0,8] $$

Let \(P\) be the COM of a system of two weights with masses \(m_{1}\) and \(m_{2}\) separated by a distance \(d\). Prove Archimedes's Law of the (weightless) Lever: \(P\) is the point on a line between the two weights such that \(m_{1} L_{1}=m_{2} L_{2},\) where \(L_{j}\) is the distance from mass \(j\) to \(P\).