Moderate -0.3 This is a straightforward volumes of revolution question requiring the standard formula V = π∫y² dx with clear bounds. The integration involves a simple power of (2x+1) which is routine with substitution or recognition, and the question explicitly provides all necessary information including the diagram. It's slightly easier than average because it's a direct application of a standard technique with no conceptual complications.
2
\includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-2_490_713_447_660}
The diagram shows part of the curve \(y = \frac { 6 } { ( 2 x + 1 ) ^ { 2 } }\). The shaded region is bounded by the curve and the lines \(x = 0 , x = 1\) and \(y = 0\). Find the exact volume of the solid produced when this shaded region is rotated completely about the \(x\)-axis.
State volume is \(\int \frac{36\pi}{(2x+1)^4}\,dx\)
B1
Or equiv in terms of \(x\); no need for limits; condone absence of \(dx\); condone absence of \(\pi\) here if it appears later
Obtain integral of form \(k(2x+1)^n\)
M1
For any \(n \leq -1\); with or without \(\pi\); or \(ku^n\) following substitution; allow if \(n=-5\); allow M1 if one slight slip occurs in \((2x+1)\)
Obtain \(-6\pi(2x+1)^{-3}\) or \(-6(2x+1)^{-3}\)
A1
Or (unsimplified) equiv
Substitute correct limits and subtract
M1
The correct way round; allow if one slight slip occurs in \((2x+1)\); not earned if limit 0 leads to \(\ldots - 0\)
Obtain \(\frac{52}{9}\pi\)
A1
Or similarly simplified exact equiv
# Question 2:
| Answer | Marks | Guidance |
|--------|-------|----------|
| State volume is $\int \frac{36\pi}{(2x+1)^4}\,dx$ | B1 | Or equiv in terms of $x$; no need for limits; condone absence of $dx$; condone absence of $\pi$ here if it appears later |
| Obtain integral of form $k(2x+1)^n$ | M1 | For any $n \leq -1$; with or without $\pi$; or $ku^n$ following substitution; allow if $n=-5$; allow M1 if one slight slip occurs in $(2x+1)$ |
| Obtain $-6\pi(2x+1)^{-3}$ or $-6(2x+1)^{-3}$ | A1 | Or (unsimplified) equiv |
| Substitute correct limits and subtract | M1 | The correct way round; allow if one slight slip occurs in $(2x+1)$; not earned if limit 0 leads to $\ldots - 0$ |
| Obtain $\frac{52}{9}\pi$ | A1 | Or similarly simplified exact equiv |
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\includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-2_490_713_447_660}
The diagram shows part of the curve $y = \frac { 6 } { ( 2 x + 1 ) ^ { 2 } }$. The shaded region is bounded by the curve and the lines $x = 0 , x = 1$ and $y = 0$. Find the exact volume of the solid produced when this shaded region is rotated completely about the $x$-axis.
\hfill \mbox{\textit{OCR C3 2012 Q2 [5]}}