Standard +0.8 This question requires students to recognize the relationship sin²x + cos²x = 1 to deduce the area of S from the area of R without repeating the trapezium rule calculation. While the trapezium rule itself is routine C2 content, the insight needed in part (iii) to exploit the Pythagorean identity and geometric relationship between the two regions elevates this above a standard exercise, requiring problem-solving rather than just procedural application.
6. (i) Write down the exact value of \(\cos \frac { \pi } { 6 }\).
The finite region \(R\) is bounded by the curve \(y = \cos ^ { 2 } x\), where \(x\) is measured in radians, the positive coordinate axes and the line \(x = \frac { \pi } { 3 }\).
(ii) Use the trapezium rule with two intervals of equal width to estimate the area of \(R\), giving your answer to 3 significant figures.
The finite region \(S\) is bounded by the curve \(y = \sin ^ { 2 } x\), where \(x\) is measured in radians, the positive coordinate axes and the line \(x = \frac { \pi } { 3 }\).
(iii) Using your answer to part (b), find an estimate for the area of \(S\).
6. (i) Write down the exact value of $\cos \frac { \pi } { 6 }$.
The finite region $R$ is bounded by the curve $y = \cos ^ { 2 } x$, where $x$ is measured in radians, the positive coordinate axes and the line $x = \frac { \pi } { 3 }$.\\
(ii) Use the trapezium rule with two intervals of equal width to estimate the area of $R$, giving your answer to 3 significant figures.
The finite region $S$ is bounded by the curve $y = \sin ^ { 2 } x$, where $x$ is measured in radians, the positive coordinate axes and the line $x = \frac { \pi } { 3 }$.\\
(iii) Using your answer to part (b), find an estimate for the area of $S$.\\
\hfill \mbox{\textit{OCR C2 Q6 [8]}}