Show that $\displaystyle\int_{0}^{\frac{\pi }{2}}{{{e}^{2y}}\cos 2y}\mathrm{ d}y=a\left( {{e}^{\pi }

Question

Show that $\displaystyle\int_{0}^{\frac{\pi }{2}}{{{e}^{2y}}\cos 2y}\mathrm{ d}y=a\left( {{e}^{\pi }}+1 \right)$ , where $a$ is a constant to be determined. [5] A curved container has a flat circular top. The shape of the container is formed by rotating part of the curve $x={{e}^{y}}\sin y$ between the point $\left( 0,0 \right)$ and $\left( {{e}^{\frac{\pi }{2}}},\frac{\pi }{2} \right)$ through $2\pi $ radians about the $y-$ axis. Find the exact volume of the container. [5]

Accepted Answer

(a) $-\frac14$; (b) $\frac\pi8(3e^\pi-1)\text{ units}^3$

2023 SAJC BT P2 Q3

2023 SAJC BT P2 Q3

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