Take the infinite limit and use the bowl radius · source reference

Printed question

(a) C: $y=e^{-x}/(1+e^{-x})$. R is bounded by C, positive $x$-axis and positive $y$-axis as shown; graph labels $y=1,y=0,1/2,R,O$. (i) Obtain a formula in terms of $n$ for $\int_0^n\frac{e^{-x}}{1+e^{-x}}dx$, $n>0$. [2] (ii) Hence find R’s exact area. [2] (iii) Denoting the answer obtained in part (i) by $q$, write down in terms of $q$ the area of R if C is scaled by factor 2 parallel to $y$-axis. [1]

(b) A girl designs a bowl by rotating section AB of $y=\ln x$ completely about $y$-axis. It must hold $300\text{ cm}^3$. A is $(a,\ln a)$; the rim diameter is twice the base diameter. State B in terms of $a$ and find exact $a$. [5]

[2010/VJC/Prelim/I/11 (modified)]

Printed working

(a)(i)

$$\int_0^n\frac{e^{-x}}{1+e^{-x}}dx=-\int_0^n\frac{-e^{-x}}{1+e^{-x}}dx=-[\ln(1+e^{-x})]_0^n=-\ln(1+e^{-n})+\ln2.$$

(a)(ii)

$$\text{Area R}=\int_0^\infty\frac{e^{-x}}{1+e^{-x}}dx=\lim_{n\to\infty}\int_0^n\frac{e^{-x}}{1+e^{-x}}dx=\lim_{n\to\infty}[-\ln(1+e^{-n})+\ln2]=-0+\ln2=\ln2,$$

since $e^{-n}\to0$ as $n\to\infty$.

(a)(iii)

When scaled by factor 2 parallel to $y$-axis, $y=2e^{-x}/(1+e^{-x})$. Required area $=\int_0^\infty2e^{-x}/(1+e^{-x})dx=2q$ as printed.

(b)

B is $(2a,\ln(2a))$.

$$\begin{aligned}\pi\int_{\ln a}^{\ln(2a)}(e^y)^2dy&=300\\\pi[\tfrac12e^{2y}]_{\ln a}^{\ln(2a)}&=300\\e^{2\ln(2a)}-e^{2\ln a}&=600/\pi\\e^{\ln((2a)^2)}-e^{\ln(a^2)}&=600/\pi\\(2a)^2-a^2&=600/\pi\\a&=\sqrt{200/\pi}\quad(a>0).\end{aligned}$$

Teacher notes

$\int f'/f=\ln|f|+c$, with the modulus sign. $|f(x)|=f(x)$ when $f(x)>0$. A scale factor 2 parallel to $y$-axis means replace $y$ by $y/2$.

Printed answer key

(a)(i) $-\ln(1+e^{-n})+\ln2$; (ii) $\ln2$; (iii) $2q$. (b) $(2a,\ln(2a))$; $a=\sqrt{200/\pi}$.

Source diagrams in native form

Mathematical review

The question defines $q$ as the finite-$n$ answer in (i), but the printed $2q$ for the entire region uses $q=\ln2$ from (ii). Correctly, finite-$n$ area doubles to $2q(n)$, while the full region’s doubled area is $2\ln2$. This reference inconsistency is qualified without rewriting the printed question.