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 \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]

$\int_0^h xy^2 dx = \int_0^h x(rx/h)^2 dx$ | M1 |
$= ... = r^2 h^3/4$ (Q) | A1 |
$\bar{x} = Q/V = 3h/4$ | M1A1 |
**OR** $(1/3)\pi rK^2 h^3\bar{x} = \int_0^h \pi rK^2 x^3 dx$ | M1M1 |
$(1/3)h^3\bar{x} = [x^4/4]_0^h$ | A1 |
$\bar{x} = 3h/4$ (AG) | A1 |

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 $\frac { 1 } { 3 } \pi r ^ { 2 } h$.]

\hfill \mbox{\textit{CAIE FP1 2008 Q1 [4]}}