What are volumes of revolution and how are they calculated?


Volumes of revolution are contructed by a fuction of x, y=f(x), rotated 360 degrees, or 2pi radians around the x asxis. For example, a function of y=2 would create a cylinder of radius 2 in its volume of revolution, and the function y=x would create a cone in its volume of revolution. This can be seen by visualising roatating to function in and out of the page using the x-axis as a centre. As the x-axis is the centre of rotation, the radius of the volume of revolution at the point will simply be equal to the value of f(x), the cross sectional area at that point will be A=pir2, or A=pif(x)2. The sum of all these areas between two x-vales should give the total volume of revolution between those two x values, this continuous sum can be defined as:


integral between (x1,x0) of pi*f(x) dx.

For example, considering a cone of radius r and height h

x1=h and  x0=0, f(0)=0 and f(h)=r, therefore f(x)=rx/h

Therefore

volume = integral between (h,0) of pi*r2x2/hdx

=0[pir2x3/3h2]h=pir2h/3

This is the correct formula for the Volume of a cone, showing the integration method works.

JA
Answered by Jack A. Maths tutor

7562 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

The polynomial f(x) is defined by f(x) = 18x^3 + 3x^2 + 28x + 12. Use the Factor Theorem to show that (3x+2) is a factor of f(x).


Find the gradient of y=x^2-6x-16 at the point where the curve crosses the x-axis


Find the turning points of the curve y=2x^3 - 3x^2 - 14.


Differentiate y= 2^x


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

MyTutor is part of the IXL family of brands:

© 2025 by IXL Learning