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Definite Integral best method to double check answers?

A problem I am dealing with is finding the Definite Integral, from 1 to 2, of (4x^3 – 3x^2) dx:

I understand that you integrate, Plug in upper limit then subtract lower limit. my answer comes out to f(x) dx = 8 is this correct? 

 

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David U. | Dave the Math TutorDave the Math Tutor
4.9 4.9 (227 lesson ratings) (227)
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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.

Maurizio T. | Statistics Ph.D and CFA charterholder with a true passion to teach.Statistics Ph.D and CFA charterholder wi...
5.0 5.0 (310 lesson ratings) (310)
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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.