Posted by **Tabby** on Sunday, April 29, 2012 at 4:05pm.

Evaluate the definite integral of

(4x^3-x^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.

- Help W/ Definite Integrals -
**Reiny**, Sunday, April 29, 2012 at 4:52pm
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/function-graphs/

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)

- Help W/ Definite Integrals -
**Tabby**, Sunday, April 29, 2012 at 5:32pm
Oh wow. Thank you so much. That was much more help than the book gave me! Thanks and God bless, Reiny!

- Help W/ Definite Integrals -
**Reiny**, Sunday, April 29, 2012 at 6:04pm
You are welcome

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