# Volume of solid of revolution

• Difficulty Level : Hard
• Last Updated : 20 Jun, 2020

A solid of revolution is generated by revolving a plane area R about a line L known as axis of revolution in the plane. Below image shows an example of solid of revolution.

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We shall calculate the volume of solid of revolution when the equation of the curve is given in parametric form and polar form.

1. Parametric Form :
If the equation of curve in parametric form is given by:
x= f(t) and y= g(t)

Where t varies from t1to t2, then the volume of revolution:

2. Polar form:
Given the equation of curve in polar form as r=f(θ), where θ varies from θ1 to θ2, the volume of revolution is calculated using the given formulas:
• About the initial line OX i.e., x-axis (θ=0) –
• About the line perpendicular to the initial line i.e. along OY (θ=π/2) –

Let us see the following examples.

Example-1:
Determine the volume of solid of revolution generated by revolving the curve whose parametric equations are –

X= 2t+3 and y= 4t2-9

About x-axis for t= -3/2 to 3/2.

Explanation :
We know that volume of solid revolved about x-axis when equation is in parametric form is given by

Using this value we get

Example-2:
Find the volume of solid generated by revolving curve r= 2a cos θ about the initial line OX.

Explanation :
We know that volume of solid generated by revolving about OX in when equation is in polar form is given by

Also for OX, θ=0
So

Put

Using these values we get

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