# Homework Help: Volumes Of Revolution

1. Jul 10, 2011

### 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.

#### Attached Files:

• ###### Forumula.jpg
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2. Jul 10, 2011

### 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!

3. Jul 10, 2011

### SammyS

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

4. Jul 10, 2011

### 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.

#### Attached Files:

• ###### trap.JPG
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5. Jul 10, 2011

### SammyS

Staff Emeritus
Actually, that should be,

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

6. Jul 11, 2011

### Endorser

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

7. Jul 11, 2011

### SammyS

Staff Emeritus
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.

8. Jul 11, 2011

### 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)