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Chapter 6. Improper integrals

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    148368
  • Chapter 6. Improper integrals

    6.1 Integrating a function that blows up

    When we integrate a function that blows up to infinity at some point in the interval we're integrating, the result may be either finite or infinite.

    Example 1
    ◊ Integrate the function y=1/√x from x=0 to x=1.

    ◊ The function blows up to infinity at one end of the region of integration, but let's just try evaluating it, and see what happens.

    eq_0866ce82.png

    eq_56022708.png

    The result turns out to be finite. Intuitively, the reason for this is that the spike at x=0 is very skinny, and gets skinny fast as we go higher and higher up.

    improper-a.jpg

    a / The integral \int01 dx/√x is finite.

    Example 2

    ◊ Integrate the function y=1/x2 from x=0 to x=1.

    eq_6222d9f9.png

    eq_5c30ce42.png

    Division by zero is undefined, so the result is undefined.

    Another way of putting it, using the hyperreal number system, is that if we were to integrate from ε to 1, where ε was an infinitesimal number, then the result would be -1+1/ε, which is infinite. The smaller we make ε, the bigger the infinite result we get out.

    Intuitively, the reason that this integral comes out infinite is that the spike at x=0 is fat, and doesn't get skinny fast enough.

    improper-b.jpg

    b / The integral \int01 dx/x2 is infinite.

    These two examples were examples of improper integrals.