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8.4 day one. Improper Integrals. Greg Kelly, Hanford High School, Richland, Washington. Sometimes we can find integrals for functions where the function or the limits are infinite. These are called improper integrals .

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8.4 day one

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8 4 day one

8.4 day one

Improper Integrals

Greg Kelly, Hanford High School, Richland, Washington


8 4 day one

Sometimes we can find integrals for functions where the function or the limits are infinite. These are called improper integrals.

Until now we have been finding integrals of continuous functions over closed intervals.


8 4 day one

Example 1:

The function is undefined at x = 1 .

Can we find the area under an infinitely high curve?

Since x = 1 is an asymptote, the function has no maximum.

We could define this integral as:

(left hand limit)

We must approach the limit from inside the interval.


8 4 day one

Rationalize the numerator.


8 4 day one

This integral converges because it approaches a solution.


8 4 day one

Example 2:

(right hand limit)

We approach the limit from inside the interval.

This integral diverges.


8 4 day one

The function approaches

when .

Example 3:


8 4 day one

If then gets bigger and bigger as , therefore the integral diverges.

If then b has a negative exponent and ,

therefore the integral converges.

Example 4:

(P is a constant.)

What happens here?

p


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