The curve $C$ has equation $$x=(y+1)e^{-\frac{y}{2}},\qquad y\ge0.$$ The line $x=1$ intersects \

Question

The curve $C$ has equation $$x=(y+1)e^{-\frac{y}{2}},\qquad y\ge0.$$ The line $x=1$ intersects $C$ at $A(1,0)$ and at a second point $B$. Using differentiation, find the exact coordinates of the point on $C$ where $x$ has its greatest value.[2] Sketch $C$ and the line $x=1$ on the same axes. State the equation of the asymptote of $C$ and find the coordinates of $B$.[3] Find the area of the finite region enclosed by $C$ and the line $x=1$.[2] Find the volume generated when this region is rotated through $2\pi$ radians about the $y$-axis.[2]

Accepted Answer

(i) $\left(2e^{-1/2},1\right)$ (ii) $x=0$, $B=(1,2.51)$ (iii) $0.348$ square units (iv) $2.37$ cubic units

An enclosed region beside a sideways exponential curve

An enclosed region beside a sideways exponential curve

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