Parametric Equation for Surface Area of Revolution

[tandoorichicken]

I know that the equation for the surface area of any solid of revolution around, say, the x-axis is
$$SA = 2\pi\int_{a}^{b} y\sqrt{1 + (\frac{\,dy}{\,dx})^2} \,dx$$

What I need is the same formula except in parametric terms, like if the problem was given in terms of x(t) and y(t). Any takers?

 

[KLscilevothma]

If it revolves about the x-axis on the closed interval [a,b], then
$$SA = 2\pi\int_{a}^{b} y(t)\sqrt{[x'(t)]^2 + [y'(t)]^2} \,dt$$

For example. if the surface are of the solid generated by revolving the region enclosed by the curve with parametric equations x(t), y(t) from t = 0 to t = pi/2, then the upper limit, b = pi/2, lower limit a = 0.

If it revolves about the y-axis on the closed interval [a,b], then
$$SA = 2\pi\int_{a}^{b} x(t)\sqrt{[x'(t)]^2 + [y'(t)]^2} \,dt$$

 

[M98Ranger]

The parametric equation for surface area of revolution can be written as:
SA = 2\pi\int_{t_1}^{t_2} y(t)\sqrt{\left(\frac{\,dx}{\,dt}\right)^2 + \left(\frac{\,dy}{\,dt}\right)^2}\,dt

This equation takes into account the changing values of x and y as t varies, and calculates the surface area by integrating over the specified range of t. It is important to note that the limits of integration, t1 and t2, will depend on the parametric equations used to describe the curve being rotated.