Mathematics 9709 · AS & A Level · Integration

Integration — practice question

The sketch displays the curve given by $y = \sqrt{\sin 2x + \sin^2 2x}$ for $0 \leq x \leq \frac{1}{6}\pi$. The shaded area lies between the curve, the vertical line $x = \frac{1}{6}\pi$ and the horizontal line $y = 0$.
(a)[3]

Apply the trapezium rule with two intervals to estimate the area of the shaded region. Give your answer correct to $2$ significant figures.

(b)[6]

The shaded region is turned completely about the $x$-axis. Find the exact volume of the solid formed.

Worked solution & mark scheme

This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Take $y$-values $(0)$, $\sqrt{\sin \frac{\pi}{6} + \sin^2 \frac{\pi}{6}}$, $\sqrt{\sin \frac{\pi}{3} + \sin^2 \frac{\pi}{3}}$ or decimal equivalents” …

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