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Definite Integral best method to double check answers?

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2 Answers

The number 8 is the correct number, and so is the process you described for how you got it.  However, what the 8 represents is not f(x)dx, but what you stated in your question, namely, the definite integral from 1 to 2 of (4x3 - 3x2).  So if you answered the question by saying "f(x)dx = 8", you could very well be marked wrong.  Labels are important!  You not only want to get something, you want to know what it is that you've got.

Yes, your answer is correct.

You compute first the indefinite integral, F(x),  for your function, f(x) = 4x3 -3x2.

In this case you get  F(x) = ∫ (4x3-3x2) dx = 4x4/4 -3x3/3+ c = x- x3 + c.

(You can verify that this is the right indefinite integral by computing the first derivative).

Now, you have to calculate F(2)-F(1) = 24-23+c -(14-13+c) = 16-8+c -(1-1+c) = 8 i.e., you have shown that ∫12 4x3 -3x2 dx = 8.