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Untitled Page 27

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    148369
  • 6.2 Limits of integration at infinity

    Another type of improper integral is one in which one of the limits of integration is infinite. The notation

    eq_496c82be.png

    means the limit of \intaH f(x)dx, where H is made to grow bigger and bigger. Alternatively, we can think of it as an integral in which the top end of the interval of integration is an infinite hyperreal number. A similar interpretation applies when the lower limit is -∞, or when both limits are infinite.

    Example 3

    ◊ Evaluate

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    eq_9afaf999.png

    eq_b3db91ee.png

    As H gets bigger and bigger, the result gets closer and closer to 1, so the result of the improper integral is 1.

    Note that this is the same graph as in example 75, but with the x and y axes interchanged; this shows that the two different types of improper integrals really aren't so different.

    improper-c.jpg

    c / The integral \int1^∞ dx/x2 is finite.

    Example 4

    ◊ Newton's law of gravity states that the gravitational force between two objects is given by F=Gm1m2/r2, where G is a constant, m1 and m2 are the objects' masses, and r is the center-to-center distance between them. Compute the work that must be done to take an object from the earth's surface, at r=a, and remove it to r=∞.

    eq_fe1d13ce.png

    eq_d463e97a.png

    eq_bb5ff191.png

    eq_b77ad777.png

    The answer is inversely proportional to a. In other words, if we were able to start from higher up, less work would have to be done.