Mathematics  Area of the surface of solid of revolution
Consider a plane y=f(x) in the xy 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 xaxis is

Area of revolution by revolving the curve about y axis is

Area of solid formed by revolving the arc of curve about xaxis is

Parametric Form:

About xaxis:

About yaxis:

About xaxis:

Polar Form: r=f(θ)

About the xaxis: initial line
Here replace r by f(θ) 
About the yaxis:
Here replace r by f(θ)

About the xaxis: initial line

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.

Limits for x: x = a to x = b
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Example:
Find the area of the solid of revolution generated by revolving the parabola about the xaxis.
Explanation:
Now we are given with the Cartesian form of the equation of parabola and the parabola has been rotated about the xaxis. Hence we use the formula for revolving Cartesian form about xaxis 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:
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