A curve $C$ has parametric equations $x=a{{\sin }^{2}}t$, $y=a\cos t$, where $0\le t\le \frac{\pi }{

Question

A curve $C$ has parametric equations $x=a{{\sin }^{2}}t$, $y=a\cos t$, where $0\le t\le \frac{\pi }{2}$ and $a>0$. Find the cartesian equation of $C$, stating clearly any restrictions on the values of $x$ and $y$. [2] Sketch $C$, showing clearly the axial intercepts. [1] The region bounded by $C$, the line $y+x=\frac{5}{4}a$ and the $y$-axis is rotated through $2\pi $ radians about the $y$-axis. Show that the exact volume of the solid obtained is $k\pi {{a}^{3}}$ where $k$ is a constant to be determined. [5]

Accepted Answer

(i) $y^{2} = a(a-x)$, $0 \leq x \leq a$, $0 \leq y \leq a$; (iii) $k = \dfrac{29}{960}$

2019 ASRJC P1 Q5

2019 ASRJC P1 Q5

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