Standard +0.3 This is a straightforward area-between-curves question requiring integration of standard functions (polynomial and exponential). Students must set up the integral correctly, integrate (1-2x)^5 using substitution or chain rule reversal and e^(2x-1) using standard exponential rules, then evaluate between 0 and 1/2. While it requires careful algebraic manipulation and knowledge of integration techniques, it follows a standard template with no novel problem-solving required, making it slightly easier than average.
5
\includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-2_486_746_1978_696}
The diagram shows the curves \(y = ( 1 - 2 x ) ^ { 5 }\) and \(y = \mathrm { e } ^ { 2 x - 1 } - 1\). The curves meet at the point \(\left( \frac { 1 } { 2 } , 0 \right)\). Find the exact area of the region (shaded in the diagram) bounded by the \(y\)-axis and by part of each curve.
At any stage of solution; if process involves two definite integrals, second must be negative
Obtain \(\frac{1}{12} + \frac{1}{2}e^{-1}\)
A1
8 Or exact equiv; no \(+c\)
## Question 5:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain integral of form $k(1-2x)^6$ | M1 | Any non-zero constant $k$ |
| Obtain correct $-\frac{1}{12}(1-2x)^6$ | A1 | Or unsimplified equiv; allow $+c$ |
| Use limits to obtain $\frac{1}{12}$ | A1 | Or exact (unsimplified) equiv |
| Obtain integral of form $ke^{2x-1}$ | M1 | Or equiv; any non-zero constant $k$ |
| Obtain correct $\frac{1}{2}e^{2x-1} - x$ | A1 | Or equiv; allow $+c$ |
| Use limits to obtain $-\frac{1}{2}e^{-1}$ | A1 | Or exact (unsimplified) equiv |
| Show correct process for finding required area | M1 | At any stage of solution; if process involves two definite integrals, second must be negative |
| Obtain $\frac{1}{12} + \frac{1}{2}e^{-1}$ | A1 | **8** Or exact equiv; no $+c$ |
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5\\
\includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-2_486_746_1978_696}
The diagram shows the curves $y = ( 1 - 2 x ) ^ { 5 }$ and $y = \mathrm { e } ^ { 2 x - 1 } - 1$. The curves meet at the point $\left( \frac { 1 } { 2 } , 0 \right)$. Find the exact area of the region (shaded in the diagram) bounded by the $y$-axis and by part of each curve.
\hfill \mbox{\textit{OCR C3 2006 Q5 [8]}}