Help W/ Definite Integrals
posted by Tabby on .
Evaluate the definite integral of
(4x^3x^2+2) from 5 to 2.
Can someone please explain what I'm being asked to do? My book is not clear on this concept.

First of all, the integral and the derivative are inverse operations
e.g.
if y = 4x^3
then dy/dx = 12x^2
then ∫ 12x^2 dx = 4x^3
notice the notation ,
read it as "the integral of 12x^2 by dx "
if we want the definite integral then usually you will find two numbers with the ∫ integral sign, the smaller number below it, and a larger number above it. (I can't type it here)
You would then substitute, and get
(value of the integral using the upper value)  (value of the integral using the lower value)
so for your question.....
∫ ( 4x^3  x^2 + 2) dx from x = 2 to 5
= [ x^4  (1/3)x^3 + 2x] from 2 to 5
= (5^4  (1/3)5^3 + 2(5) )  (2^4  (1/3)(2^3) + 2(2) )
= 625  125/3 + 10  16 + 8/3  4
= 615  117/3
= 615 39
= 576
(check my arithmetic, I am prone to errors today)
To see a geometrical interpretation of what we did, do the following
go to
http://rechneronline.de/functiongraphs/
in the "first graph" window enter:
4x^3  x^2 + 2 , (type it exactly that way)
in 'Range xaxis from' enter 2 and 5
in 'Range yaxis from' enter 0 and 500
click on "Draw"
What our answer of 576 represents is the area between the curve and the xaxis from x = 2 to x = 5
(our answer of 576 is reasonable if we consider the
average height of the "triangle" to be (477+30)/2= appr 254 and our base is 3, from 2 to 5
254x3 = 762
of course this answer is too large, since we joined the endpoints with a straight line) 
Oh wow. Thank you so much. That was much more help than the book gave me! Thanks and God bless, Reiny!

You are welcome