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Example 10

Find

06_applications_of_the_integral-456.gif.

We see that

06_applications_of_the_integral-457.gif

and

06_applications_of_the_integral-458.gif

Thus 06_applications_of_the_integral-459.gif diverges and has the form ∞ - ∞. We do not assign it any value or either of the symbols ∞ or -∞. The region under the curve f(x) = x is shown in Figure 6.7.10.

06_applications_of_the_integral-460.gif

Figure 6.7.10

It is tempting to argue that the positive area to the right of the origin and the negative area to the left exactly cancel each other out so that the improper integral is zero. But this leads to a paradox.

Wrong:

06_applications_of_the_integral-461.gif

Let

v = x + 2, dv = dx.

Then

06_applications_of_the_integral-462.gif

Subtracting

06_applications_of_the_integral-463.gif

But

06_applications_of_the_integral-464.gif

So we do not give the integral 06_applications_of_the_integral-465.gif the value 0, and instead leave it undefined.


Last Update: 2006-11-15