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Math Help - Finding area enclosed by integral and x-axis

  1. #1
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    Finding area enclosed by integral and x-axis

    Find the area enclosed by the graph of definite integral [4(x^3-3x^2+2x)dx] (a=0, b=2) and the x-axis. The first part of this question asked to evaluate the definite integral for which i got 0 and i thought you would find the area the same way but the last part asks you to explain why the two answers aren't the same!
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  2. #2
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    If you graph the equation, you will see that an equal portion lies in the negative y axis and the positive. When you run the integral from 0 to 1 and 1 to 2, you will get the area correct.Finding area enclosed by integral and x-axis-graph.pdf

    See graph
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  3. #3
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    Note that area,whether above or below the axis, is always positive.

    Integrating \int_a^b f(x)dx will give the area between the graph of f(x) and the x-axis only if f(x) is non-negative between a and b.

    By the way, your phrasing is rather peculiar. This is not the area between an integral and the x-axis, it is the area between a graph and the x-axis. And not the "graph of the definite integral". A definite integral is always a number and has no "graph". You are asking for the area between the graph of y= 4(x^3-3x^2+2x) and the x-axis and you find that by taking the definite integral (and being careful of the sign as I say).
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