Mathematics | Area of the surface of solid of revolution

Difficulty Level : Medium
Last Updated : 10 May, 2020

Consider a plane y=f(x) in the x-y plane between ordinates x=a and x=b. If a certain portion of this curve is revolved about an axis, a solid of revolution is generated.

We can calculate the area of this revolution in various ways such as:

Cartesian Form:
- Area of solid formed by revolving the arc of curve about x-axis is-
  $S= \int_{x=a}^{x=b} 2\pi y\sqrt{1+(\frac{dy}{dx})^2}dx$
- Area of revolution by revolving the curve about y axis is-
  $S= \int_{y=c}^{y=d} 2\pi x \sqrt{1+(\frac{dx}{dy})^2}dy$

Parametric Form: $x=x(t), y=y(t)$
- About x-axis:
  $S=\int_{t=t_{1}}^{t=t_{2}} 2\pi y(t) \sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}dt$
- About y-axis:
  $S=\int_{t=t_{1}}^{t=t_{2}} 2\pi x(t) \sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}dt$

Polar Form: r=f(θ)
- About the x-axis: initial line $\theta = \frac{\pi}{2}$
  $S= \int_{\theta=\theta_1}^{\theta _2}2\pi y\frac{ds}{d\theta}d\theta$
  $=\int_{\theta=\theta_1}^{\theta _2}2\pi (r sin\theta) \sqrt{r^2+(\frac {dr}{d\theta})^2}d\theta$
  Here replace r by f(θ)
- About the y-axis:
  $S= \int_{\theta=\theta_1}^{\theta _2}2\pi x\frac{ds}{d\theta}d\theta$
  $=\int_{\theta=\theta_1}^{\theta _2}2\pi (r cos\theta) \sqrt{r^2+(\frac {dr}{d\theta})^2}d\theta$
  Here replace r by f(θ)

About any axis or line L: $S= \int 2\pi (PM) ds$ where PM is the perpendicular distance of a point P of the curve to the given axis.
- Limits for x: x = a to x = b
  $S=\int_{x=a}^{x=b} 2\pi (PM)\sqrt{1+(\frac{dy}{dx})^2}dx$
  Here PM is in terms of x.
- Limits for y: y = c to y = d
  $S= \int_{y=c}^{y=d} 2\pi (PM)\sqrt{1+(\frac{dx}{dy})^2}dy$
  Here PM is in terms of y.

Example:
Find the area of the solid of revolution generated by revolving the parabola $y^2=4ax, 0\leq x \leq 3a$ about the x-axis.
Explanation:
Now we are given with the Cartesian form of the equation of parabola and the parabola has been rotated about the x-axis. Hence we use the formula for revolving Cartesian form about x-axis which is: