On calculator papers, use a graph to locate the required region, solve the intersection equation numerically, then evaluate the correct definite integral. Keep the set-up visible in your written solution.
| Step | What to record |
|---|---|
| Locate | A sketch with the relevant curves, intersections and shaded region. |
| Solve | The equation \(\mathrm{f}(x)=\mathrm{g}(x)\) and the required roots. |
| Integrate | An upper-minus-lower integral for each interval. |
| Report | The area, units and requested accuracy. Retain full precision until the final line. |
A curve and a chord can both lie below the horizontal axis. The region between them still uses upper minus lower; do not automatically negate each function separately.
For \(y=\sin x\) and \(y=\dfrac{x}{2}\), the positive non-zero intersection is \(\alpha\approx1.89549\) (radians).
\(A=\int_0^\alpha\left(\sin x-\dfrac{x}{2}\right)\,\mathrm{d}x=\left[-\cos x-\dfrac{x^2}{4}\right]_0^\alpha\approx0.421\).
Use radian mode for trigonometric calculus. For “show that” commands, display the algebra or reasoning; a numerical value alone is insufficient.
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