Calculus

Find the exact area below the curve (1-x)*x^9 and above the x axis

asked by Amanda
  1. Were you given (a,b) values?

    From x = ? to ?

    posted by helper
  2. First you define the function as
    f(x)=(1-x)*x^9

    Inspection of its 10 factors should indicate that the function crosses the x-axis at two points (0,0), (0,1).

    The leading coefficient is -x^10, which means that the major part of the function is concave downwards, and is above the x-axis only between x=0 and 1.

    Check: f(0.5)=(1-0.5)*0.5^9 > 0

    The area sought is thus between the limits of x=0 and x=1.

    The area below a curve f(x) is
    ∫f(x)dx between the limits of integration (0 to 1).

    The function can be split up into a polynomial with two terms, and is thus easy to integrate.

    Inspection of the graph between 0 and 1 and an approximate calculation of the area shows that the area should be in the order of 0.01. Post your answer for a check if you wish.

    Here's a graph of the function between 0 and 1.

    http://img411.imageshack.us/img411/1397/1296510221.png

    posted by MathMate
  3. I just looked at the graph.

    You want the area from x = 0 to x = 1.

    | = integrate symbol
    | x^9(1 - x)
    | x^9 - x^10

    Then plug in x, from 0 to 1.

    posted by helper

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