Definite Integral best method to double check answers?

Question

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? 

 

Answer 1

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.

Answer 2

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 = x⁴ - 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.