Friday, April 28, 2006

Integration: A Problem of Decimals

Integration is a pain in the ass. Just as much so for third graders is division.

Multiplication is simple as long as one can understand addition. 5*3 = 5 + 5 + 5 = 15, or take 3 instances of 5 and add them together.

Division is backasswards. 30/5 is not so simply an action, it is an inversal. This process is typically simplified though and is comprehended as "what multiplies by 5 to equal 30." But the true meaning of this division is to figure out, if you were to split 30 into five parts, how large those parts would each be. The typical simplification turns 30/5 = x into 5*x=30, and asks for it to be solved intuitively. Using this system, one must know the multiplication tables to have any hope with division.

Integration and derivation are the same way. Inorder to integrate, one must have a strong knowledge of derivatives. When integrating, however, one is trained to think "hmm, what derives to equal this." On the contrary, the thinking should be entirely different: "hmm, what formula would best represent another dimension* of this function."

*I have another explanation for this but don't feel like explaining right now.

Consider thus. Say, for example,
37/5 = 7 + 2/5
A mixed number. This is how we represent integrals.

Why not give a "decimal" representation of integrals? How would this work?


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