Simple Rules for Differentiation



Infinite Intervals of Integration

Objectives:

Students will be able to

• Use limits to determine if infinite intervals of integration are convergent or divergent.

• Calculate improper integrals (where possible).

Here we are going to examine how to find calculate a definite integral where one of the limits of integration is infinite. Integrals that have one or more infinite limits of integration are called improper integrals. In order to calculate improper integrals, we will once again use limits.

• [pic]

• [pic]

• [pic]

where a, b, and c are real numbers and c is arbitrarily chosen.

When the limit of an improper integral exists, then the integral is said to be convergent. If the limit of an improper integral does not exist, then the integral is said to be divergent.

Example 1:

Find the area, if it is finite, of the region under the graph of [pic] over the interval [pic].

Example 2:

Determine whether the improper integral [pic] converges or diverges. If it converges, find the value.

Example 3:

Determine whether the improper integral [pic] converges or diverges. If it converges, find the value.

Example 4:

Determine whether the improper integral [pic] converges or diverges. If it converges, find the value.

Example 5:

Find [pic], if possible.

Example 6:

An investment produces a perpetual stream of income with a flow rate of [pic]. Find the capital value at an interest rate of 10% compounded continuously.

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