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### 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
```
Associated Topics:
High School Calculus

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