The curves $C_1$ and $C_2$ have equations \[C_1:\quad x=\frac{\sqrt y}{4-y^2},\qquad C_2:\quad x

Question

The curves $C_1$ and $C_2$ have equations $$C_1:\quad x=\frac{\sqrt y}{4-y^2},\qquad C_2:\quad x=1-\frac y2,$$ where $0\le y<2$. The curves intersect at $P$. The region $R$ is bounded by $C_1$, $C_2$ and the $x$-axis. Sketch $C_1$ and $C_2$ on the same axes, indicating their intersections with the $x$-axis. Find the coordinates of $P$.[3] Find the area of $R$.[3] Find the volume generated when $R$ is rotated through $2\pi$ radians about the $y$-axis.[4]

Accepted Answer

(i) $P=(0.413,1.17)$, intercepts $(0,0)$ and $(1,0)$ (ii) $0.577$ square units (iii) $1.74$ cubic units

Area and volume between sideways curves

Area and volume between sideways curves

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