Use unsigned area and subtract the rotating core · source reference

Printed question

(a) $x=2t^2$, $y=\sin(-t)$, $t\ge0$. Find the exact area bounded by curve, $x$-axis and $x=\pi^2/8$. [5]

(b) C: $y=x^2\sec x\tan^2x$, $-\pi/2<x<\pi/2$. R is bounded by C and $y=\frac23\pi^2$. Find its volume rotating $2\pi$ about $x$-axis to 2 decimal places. [4]

[2011/MJC/Prelim/II/2]

Printed working

(a)

$x=2t^2$, $dx/dt=4t$. $x=0\Rightarrow t=0$; $x=\pi^2/8\Rightarrow t=\pi/4$ since $t\ge0$.

$$\begin{aligned}\text{Area}&=-\int_0^{\pi^2/8}y dx=-\int_0^{\pi/4}\sin(-t)(4t)dt=4\int_0^{\pi/4}t\sin t dt\\&=4[-t\cos t]_0^{\pi/4}-4\int_0^{\pi/4}(-\cos t)dt\\&=4(-\sqrt2\pi/8)+4[\sin t]_0^{\pi/4}=2\sqrt2-\pi\sqrt2/2\quad\text{units}^2.\end{aligned}$$

(b)

When $y=\frac23\pi^2$, $x=\pm\pi/3$ or $\pm1.0472$.

$$V=\pi(2\pi^2/3)^2(2\pi/3)-2\int_0^{\pi/3}\pi(x^2\sec x\tan^2x)^2dx=267.73\quad\text{units}^3\ (2\text{ d.p.}).$$

Teacher notes

The graph shades area below the $x$-axis in (a). In (b), subtract the blue even curved solid from the red cylinder: cylinder radius $2\pi^2/3$, length $2\pi/3$. Graph labels $y=f(x)$, R,O,$\pm\pi/2$ asymptotes,$\pm\pi/3$ boundaries, $y=2\pi^2/3$ and rotation arrows.

Printed answer key

(a) $2\sqrt2-\pi\sqrt2/2$ units$^2$; (b) $267.73$ units$^3$ (2 d.p.).

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