Standard +0.8 This is a volume of revolution problem requiring integration of a non-standard function. While the setup is straightforward (standard formula π∫y² dx), the integration of 64/(16+x³) requires either partial fractions with complex roots or a substitution that students must recognize. The 'exact value' requirement and the cubic denominator elevate this above routine C3/C4 volume questions, making it moderately challenging for Further Pure.
In this question you must show detailed reasoning.
The finite region \(R\) is enclosed by the curve with equation \(y = \frac{8}{\sqrt{16+x^3}}\), the \(x\)-axis and the lines \(x=0\) and \(x=4\). Region \(R\) is rotated through \(360°\) about the \(x\)-axis. Find the exact value of the volume generated. [4]
Question 2:
2 | 4 2
(cid:180) (cid:167) 8 (cid:183)
DR [V (cid:32)](cid:83)(cid:181) (cid:168) (cid:184) dx
(cid:181) (cid:168) (cid:184)
16(cid:14)x2
(cid:182) (cid:169) (cid:185)
0
(cid:167)x(cid:183)
64(cid:83)(cid:117)1tan(cid:16)1
(cid:168) (cid:184)
4 (cid:169)4(cid:185)
16(cid:83)tan(cid:16)1(cid:11)1(cid:12)(cid:16)16(cid:83)tan(cid:16)1(cid:11)0(cid:12)
(cid:32)4(cid:83)2 | B1
M1
c
eM1
A1
[4] | m
1.1
i
1.2
1.1
1.1 | e | Must be seen
Using formulae booklet
Substitution of correct limits
must be seen
In this question you must show detailed reasoning.
The finite region $R$ is enclosed by the curve with equation $y = \frac{8}{\sqrt{16+x^3}}$, the $x$-axis and the lines $x=0$ and $x=4$. Region $R$ is rotated through $360°$ about the $x$-axis. Find the exact value of the volume generated. [4]
\hfill \mbox{\textit{OCR Further Pure Core 2 Q2 [4]}}