Help W/ Definite Integrals

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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 -

    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 -

    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 -

    You are welcome

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