First of all, the integral and the derivative are inverse operations
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
(check my arithmetic, I am prone to errors today)
To see a geometrical interpretation of what we did, do the following
in the "first graph" window enter:
4x^3 - x^2 + 2 , (type it exactly that way)
in 'Range x-axis from' enter 2 and 5
in 'Range y-axis from' enter 0 and 500
click on "Draw"
What our answer of 576 represents is the area between the curve and the x-axis 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