Challenging +1.2 This is a standard M4 centre of mass problem requiring integration to find the centroid of a region between two curves. While it involves multiple steps (finding intersection points, setting up the integral for area and moment, then computing the ratio), the techniques are routine for Further Maths M4 students. The trigonometric functions and exact value requirement add moderate complexity, but the bounds are given and the setup follows a standard template.
3 The region bounded by the \(y\)-axis and the curves \(y = \sin 2 x\) and \(y = \sqrt { 2 } \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\) is occupied by a uniform lamina. Find the exact value of the \(x\)-coordinate of the centre of mass of the lamina.
# Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $A = \int_0^{\pi/4}(\sqrt{2}\cos x - \sin 2x)\,dx$ | M1* | Attempt at integration to find area (both terms including subtraction); Limits not required for M and first A mark |
| $= \left[\sqrt{2}\sin x + \frac{1}{2}\cos 2x\right]_0^{\pi/4} = \frac{1}{2}$ | A1A1 | A1 for both terms correct, A1 for $\frac{1}{2}$ |
| $A\bar{x} = \int_0^{\pi/4} x(\sqrt{2}\cos x - \sin 2x)\,dx$ | M1* | Integration by parts; Clear indication of integrating trigonometric term and differentiating $x$ term |
| $= \left[x\left(\sqrt{2}\sin x + \frac{1}{2}\cos 2x\right)\right]_0^{\pi/4} - \int_0^{\pi/4}\left(\sqrt{2}\sin x + \frac{1}{2}\cos 2x\right)dx$ | | |
| $= \left[x\left(\sqrt{2}\sin x + \frac{1}{2}\cos 2x\right) + \sqrt{2}\cos x - \frac{1}{4}\sin 2x\right]_0^{\pi/4}$ | A2 | Both terms integrated correctly (A1 for one error); Limits not required for M and A marks (for $A\bar{x}$) |
| $= \frac{\pi}{4} + \frac{3}{4} - \sqrt{2}$ | | |
| $\bar{x} = \frac{\left(\frac{\pi}{4}+\frac{3}{4}-\sqrt{2}\right)}{\left(\frac{1}{2}\right)} = \frac{\pi}{2}+\frac{3}{2}-2\sqrt{2}$ | M1 dep*, A1 | M1 for $\bar{x} = \frac{A\bar{x}}{A}$ |
| **[8]** | | |
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3 The region bounded by the $y$-axis and the curves $y = \sin 2 x$ and $y = \sqrt { 2 } \cos x$ for $0 \leqslant x \leqslant \frac { 1 } { 4 } \pi$ is occupied by a uniform lamina. Find the exact value of the $x$-coordinate of the centre of mass of the lamina.
\hfill \mbox{\textit{OCR M4 2014 Q3 [8]}}