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
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