If a curve is given in parametric form, is there any way that I can find the area bounded by the curve and the x-axis without finding $y$ as an explicit function of $x$?

Consider this curve for which $$y=a(1-\cos t)$$ and $$x=a(t-\sin t)$$

I need to find the area bounded by an arc of this curve and the x-axis.

Expressing $y$ in terms of $x$ would become a tedious job here.

The only thing I can make out is that $y=dx/dt$. Will it be of any help in this case?

Due to the parametric form, I am not able to draw an approximate graph of the function.

Can anybody help on the approach?


if your parametric curves are:

$$ x(t) = f(t)\\ y(t) = g(t)\\ t_0\le t\le t_1 $$

Then the area between the arc and the x-axis is:

$$ A = \int{ydx}=\int_{t_0}^{t_1}{g(t)f'(t)dt} $$

So, in your example,

$$ A = \int_{t_0}^{t_1}{a(1-\cos t)a(1-\cos t)dt}=a^2\left( \frac{1}{4} (6 t - 8 \sin t + \sin 2t) \right)_{t=t_0}^{t=t_1} $$


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