Integration problem, stuck at this.

Question

Here's the parametric equations:

$x = \sin (t)$ $y = \sin (2t)$

$ 0 \le t \le \frac{\pi}{2}$

(1) How would I find the area of the region bounded by the curve and the x axis. (2) Volume of the solid formed when rotated $2\pi$ radians about x axis.

Answer 1

Took me a while, but solved. Was stuck in the integration part to be specific.

Answer 2

In this case, the ordinary formulas for the area and the volume are $\int^b_a y \ dx$ and $\pi\int^b_a y^2 \ dx$.

For parametric equations, you replace $y$ with $y(t)$, $dx$ with $x'(t)dt$, and the limits of integration with the corresponding $t$ values, just as using the substitution rule for definite integrals.