Math Questions

Date: 19 Mar 1995 04:41:25 -0500
From: Nicholas Holton
Subject: math question

Why is anything multiplied by zero equal to zero?

What is the volume of the solid of revolution created by 
the function, y=cos(x) and y=(x+1)/3, when these are 
revolved about the x-axis?

Date: 21 Mar 1995 15:03:58 -0500
From: Dr. Sydney
Subject: Re: math question

Dear Nicholas,

Hello!  We are glad you wrote to Dr. Math!  

When we multiply two numbers together, like 2 and 3, we are 
really counting how much 2 3's are, or 3 2's are.  Another way 
to look at is we are adding 3 to itself 2 times; or we are adding 
2 to itself 3 times.  No matter which way we look at it, we still 
get 6 as our result.  So, if we multiply a number like 28 by zero 
that is like asking how much 0 28's are, or 28 0's are.  Or, how 
much adding 28 to itself 0 times is, or how much adding 0 to 
itself is 28 times.  The answer is going to be 0 no matter how 
we look at it.  Can you see why it holds that ANY number 
multiplied by 0 is 0?  

Another perspective on this is that anything multiplied by 0 is 0 
by definition.  You can actually rigorously prove that 0a = 0 
(where a is any real number).  Using the fact that 0 + 0 = 0, we 
can say that (0+0)a = 0a.  Now, distribute the a on the left to get 
0a + 0a = 0a.  Can you see what the last step would be to show 
that 0a = 0?

On to your other question...

Before we answer this I just want to clarify something...do you 
mean you want to revolve around the x-axis the region that is 
bounded by the functions y=cosx and y=(x+1)/3 and the x-axis?  
If you write back to confirm this, we'll help out.  

Hope this helps!  Write back if you have any more questions.

--Sydney, Dr. Math