Standard +0.3 This is a standard volumes of revolution question requiring the formula V = π∫[f(x)² - g(x)²]dx. The main steps are straightforward: identify which curve is outer/inner, set up the integral with correct limits, and integrate e^(6x) and (2x-1)^8. The substitution for (2x-1)^8 and integration of exponentials are routine C3 techniques, making this slightly easier than average.
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\includegraphics[max width=\textwidth, alt={}, center]{5c501214-b41c-43a8-b9c6-986758e83e7d-3_586_798_267_676}
The diagram shows the curves \(y = \mathrm { e } ^ { 3 x }\) and \(y = ( 2 x - 1 ) ^ { 4 }\). The shaded region is bounded by the two curves and the line \(x = \frac { 1 } { 2 }\). The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced.
Obtain correct indefinite integral of their \(k_1e^{nx}\)
A1
Substitute limits to obtain \(\frac{1}{k}e(e^3-1)\) or \(\frac{1}{k}(e^3-1)\)
A1
or exact equiv perhaps involving \(e^0\)
Integrate \(k(2x-1)^n\) to obtain \(k'(2x-1)^{n+1}\)
M1
any constants involving \(\pi\) or not; any \(n\)
Obtain correct indefinite integral of their \(k(2x-1)^n\)
A1
Substitute limits to obtain \(\frac{1}{18}\pi\) or \(\frac{1}{9}\)
A1
or exact equiv
Apply formula \(\int\pi y^2\mathrm{d}x\) at least once
B1
for \(y = e^{3x}\) and/or \(y = (2x-1)^4\)
Subtract, correct way round, attempts at volumes
M1
allow with \(\pi\) missing but must involve
Obtain \(\frac{1}{9}\pi e^3 - \frac{2}{9}\pi\)
A1
or similarly simplified exact equiv
| Integrate $k_1e^{nx}$ to obtain $k_2e^{nx}$ | M1 | any constants involving $\pi$ or not; any $n$ |
| Obtain correct indefinite integral of their $k_1e^{nx}$ | A1 | |
| Substitute limits to obtain $\frac{1}{k}e(e^3-1)$ or $\frac{1}{k}(e^3-1)$ | A1 | or exact equiv perhaps involving $e^0$ |
| Integrate $k(2x-1)^n$ to obtain $k'(2x-1)^{n+1}$ | M1 | any constants involving $\pi$ or not; any $n$ |
| Obtain correct indefinite integral of their $k(2x-1)^n$ | A1 | |
| Substitute limits to obtain $\frac{1}{18}\pi$ or $\frac{1}{9}$ | A1 | or exact equiv |
| Apply formula $\int\pi y^2\mathrm{d}x$ at least once | B1 | for $y = e^{3x}$ and/or $y = (2x-1)^4$ |
| Subtract, correct way round, attempts at volumes | M1 | allow with $\pi$ missing but must involve |
| Obtain $\frac{1}{9}\pi e^3 - \frac{2}{9}\pi$ | A1 | or similarly simplified exact equiv |
6\\
\includegraphics[max width=\textwidth, alt={}, center]{5c501214-b41c-43a8-b9c6-986758e83e7d-3_586_798_267_676}
The diagram shows the curves $y = \mathrm { e } ^ { 3 x }$ and $y = ( 2 x - 1 ) ^ { 4 }$. The shaded region is bounded by the two curves and the line $x = \frac { 1 } { 2 }$. The shaded region is rotated completely about the $x$-axis. Find the exact volume of the solid produced.
\hfill \mbox{\textit{OCR C3 2008 Q6 [9]}}