Prove standard centre of mass formula

A question is this type if and only if it asks to prove by integration a standard result for centre of mass (e.g. semicircle at 4r/3π, cone at 3h/4, hemisphere at 3r/8).

2 questions · Standard +0.8

6.04d Integration: for centre of mass of laminas/solids
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CAIE FP1 2008 June Q1
4 marks Standard +0.8
1 The finite region enclosed by the line \(y = k x\), where \(k\) is a positive constant, the \(x\)-axis for \(0 \leqslant x \leqslant h\), and the line \(x = h\) is rotated through 1 complete revolution about the \(x\)-axis. Prove by integration that the centroid of the resulting cone is at a distance \(\frac { 3 } { 4 } h\) from the origin \(O\).
[0pt] [The volume of a cone of height \(h\) and base radius \(r\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
Edexcel M3 Q5
13 marks Standard +0.8
5. (a) Use integration to show that the centre of mass of a uniform solid right circular cone of height \(h\) is \(\frac { 3 } { 4 } h\) from the vertex of the cone.
(6 marks) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ad523c3f-9109-45a8-8399-80a4c2edeff7-4_419_424_372_721} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} A paperweight is made by removing material from the top half of a solid sphere of radius \(r\) so that the remaining solid consists of a hemisphere of radius \(r\) and a cone of height \(r\) and base radius \(r\) as shown in Figure 3.
(b) Find the distance of the centre of mass of the paperweight from its vertex.
(7 marks)