Mathematics | Area of the surface of solid of revolution

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:

1. Cartesian Form:
   • Area of solid formed by revolving the arc of curve about x-axis is-
   • Area of revolution by revolving the curve about y axis is-
2. Parametric Form:
   • About x-axis:
   • About y-axis:
3. Polar Form: r=f(θ)
   • About the x-axis: initial line Here replace r by f(θ)
   • About the y-axis: Here replace r by f(θ)
4. About any axis or line L: 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 Here PM is in terms of x.
   • Limits for y: y = c to y = d Here PM is in terms of y.

Example: Find the area of the solid of revolution generated by revolving the parabola 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: Here . Now we need to calculate dy/dx Differentiating w.r.t x we get: Using Now we are provided with limits of x as x=0 to x=3. Plugging our calculated values in the above formula we get: