The curve $C\,$ has the parametric equations

\sin t \quad y = 3 \sin 2t \quad 0 \leq t \leq \frac{\pi}{2}

A first-quadrant graph with arrowed x- and y-axes, the curve C from the origin to the x-axis, and the shaded region R beneath it.

Show that $dxdt=3cos⁡t\displaystyle \frac{dx}{dt}=3\cos t$

Show that the area of $R\,$ is given by $dt\displaystyle \int_0^{\frac{\pi}{2}} 18 \sin t \cos^2 t\,dt$

Hence show, by algebraic integration, that the area of $R\,$ is exactly 6

The curve $C\,$ has the parametric equations

\sin t \quad y = 3 \sin 2t \quad 0 \leq t \leq \frac{\pi}{2}

A first-quadrant graph with arrowed x- and y-axes, the curve C from the origin to the x-axis, and the shaded region R beneath it.

Show that $dxdt=3cos⁡t\displaystyle \frac{dx}{dt}=3\cos t$

Show that the area of $R\,$ is given by $dt\displaystyle \int_0^{\frac{\pi}{2}} 18 \sin t \cos^2 t\,dt$

Hence show, by algebraic integration, that the area of $R\,$ is exactly 6

Integration

Integration Questions