Section 7.7: Improper Integrals. If makes sense, then what about ? Definition This is our first example of an improper integral.

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In this section, we will define what it means for an integral to be improper and begin investigating how to determine convergence or divergence of such.

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Presentation transcript:

Section 7.7: Improper Integrals

If makes sense, then what about ? Definition This is our first example of an improper integral.

If the limit is finite then we say that the improper integral converges otherwise the integral diverges. Notice that if the integral is going to converge, Zero must be a horizontal asymptote and the function must get small quickly.

Fact

Vertical Asymptotes provide another kind of improper integral. You always have to check that the function is continuous. Diverges, but if we ignore the asymptote, we get the wrong answer:

Third Type

Volume = This is a paint can that can be filled with π unit 3 of paint, but the surface requires an infinite amount of paint! Gabriel’s horn Rotate f(x) = 1/x about x-axis Surface Area = Diverges!!