1.45: Past-paper question 10

(a) Using a formula from page 2, show that

Figure 2 shows part of the curve $\displaystyle C$ with equation $\displaystyle y = \mathrm{sin} x$ and part of the curve $\displaystyle D$ with equation $\displaystyle y = \mathrm{cos} 2x$

Curve $\displaystyle C$ and curve $\displaystyle D$ intersect at the point $\displaystyle A$

(b) Use algebra to find the exact coordinates of $\displaystyle A$

The shaded region $\displaystyle R$ is rotated through $\displaystyle 360^\circ$ about the $\displaystyle x$-axis.

(c) Use algebraic integration to find the exact volume of the solid generated.

Give your answer in the form $\displaystyle \frac{a\sqrt{a}}{b}\pi$

where $\displaystyle a$ is a prime number and $\displaystyle b$ is an integer.