How can we use calculus to find volumes of revolution?

[Endorser]

As part of an assignment on Approximating Areas and Volume I am asked to derive the equation shown in the image attached.

The question reads: "It can be shown that if y = f(x) is revolved around the x-axis to form a solid between x=a and x=b then the volume of the solid is give by the equation shown in the image.

Some equations I have been using are basic area forumla such as
Area (trapezium) = 1/2(a+b)xh

I have also attempted to derive the forumula by looking at the Trapezoidal Rule and Simpson's Method and working backward to derive the formula.

 

[mattg443]

As you rotate a cross section of the curve around the axis, it forms a cylinder, with radius y=f(x). and a thickness of δx.

The volume of that cylinder is given by:
A=$\pi$y² δx

As the thickness of the cylinder approaches zero and you add (integrate) all the volumes of the reaaaaaly thin cylinders.
That gives the expression:

$\int$$\pi$f(x)²dx

I'm not quite sure how to put the limits in, but they are from a to b.

I hope that helped!

 

[SammyS]

$\displaystyle \int_{a}^{b}{\pi \left(f(x)\right)^2} dx$

 

[None_of_the]

Hi,
Use two fonctions y=mx for a to b and y=k for b to c

add the result and multiply by two.

 

[SammyS]

As you rotate a cross section of the curve around the axis, it forms a cylinder, with radius y=f(x). and a thickness of δx.

The volume of that cylinder is given by:
A=$\pi$y² δx

As the thickness of the cylinder approaches zero and you add (integrate) all the volumes of the reaaaaaly thin cylinders.
That gives the expression:

$\int$$\pi\,$f(x)²dx

I'm not quite sure how to put the limits in, but they are from a to b.

I hope that helped!

Actually, that should be,

The volume of that cylinder is given by:
A*δx = $\pi$y² δx

 

[Endorser]

Thanks All, But how to we actually get from A=πy2 δx to ∫πf(x)2dx
How does the area become the volume?

 

[SammyS]

A is the area of a circle with radius y & y = f(x). That radius goes from the x-axis, vertically up to the graph y = f(x). Multiplying times δx (delta-x) gives the volume of a very thin circular disk of thickness δx . The integral from x=a to x=b indicates that the volume of a series of such disk is summed to give the total volume of the solid of revolution.

 

[mattg443]

The definition of an integral is adding ($\sum$) very skinny things (lim$\delta$ x-->0) between two points.

If you are finding the area under a curve, you integrate between two points and are adding skinny rectangles (with almost no width i.e$\delta$x), which are practically adding lines. (you just add the heights $\int$ydx)

With volumes, in this case, you are adding skinny cylinders, until the cylinder becomes practically a circle. (you just add the areas of those circles $\int$$\pi$y²dx)

Adding lines gives an area
Adding areas gives a volume (just imagine adding all the areas of the pages of a book to get its volume)