# 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:**

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**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-**

**Parametric Form:****About x-axis:****About y-axis:**

**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)

**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: