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More on Improper Integrals Help

By — McGraw-Hill Professional
Updated on Aug 31, 2011

Introduction to More on Improper Integrals

Suppose that we want to calculate the integral of a continuous function f ( x ) over an unbounded interval of the form [ A , +∞) or (−∞, B ]. The theory of the integral that we learned earlier does not cover this situation, and some new concepts are needed. We treat improper integrals on infinite intervals in this section, and give some applications at the end.  

The Integral on an Infinite Interval

Let f be a continuous function whose domain contains an interval of the form [ A , +∞). The value of the improper integral

Indeterminate Forms 5.4 More on Improper Integrals

is defined to be

Indeterminate Forms 5.4 More on Improper Integrals

Similarly, we have: Let g be a continuous function whose domain contains an interval of the form (−∞, B ]. The value of the improper integral

Indeterminate Forms 5.4 More on Improper Integrals

is defined to be

Indeterminate Forms 5.4 More on Improper Integrals

Examples

Example 1

Calculate the improper integral

Indeterminate Forms 5.4 More on Improper Integrals

Solution 1

We do this problem by evaluating the limit

Indeterminate Forms 5.4 More on Improper Integrals

We conclude that the integral converges and has value 1/2.

Example 2

Evaluate the improper integral

Indeterminate Forms 5.4 More on Improper Integrals

Solution 2

We do this problem by evaluating the limit

Indeterminate Forms 5.4 More on Improper Integrals

This limit equals −∞. Therefore the integral diverges.

You Try It : Evaluate Indeterminate Forms 5.4 More on Improper Integrals .

Doubly Infinite Integrals

Sometimes we have occasion to evaluate a doubly infinite integral. We do so by breaking the integral up into two separate improper integrals, each of which can be evaluated with just one limit.

Example 1

Evaluate the improper integral

Indeterminate Forms 5.4 More on Improper Integrals

Solution 1

The interval of integration is (−∞, +∞). To evaluate this integral, we break the interval up into two pieces:

(−∞, +∞) = (−∞, 0] ∪ [0, +∞).

(The choice of zero as a place to break the interval is not important; any other point would do in this example.) Thus we will evaluate separately the integrals

Indeterminate Forms 5.4 More on Improper Integrals

For the first one we consider the limit

Indeterminate Forms 5.4 More on Improper Integrals

The second integral is evaluated similarly:

Indeterminate Forms 5.4 More on Improper Integrals

Since each of the integrals on the half line is convergent, we conclude that the original improper integral over the entire real line is convergent and that its value is

Indeterminate Forms 5.4 More on Improper Integrals

You Try It : Discuss Indeterminate Forms 5.4 More on Improper Integrals .

Some Applications

Now we may use improper integrals over infinite intervals to calculate area.

Examples

Example 1

Calculate the area under the curve y = 1/[ x · (lnx)4 ] and above the x -axis, 2 ≤ x < ∞.

Solution 1

The area is given by the improper integral

Indeterminate Forms 5.4 More on Improper Integrals

For clarity, we let φ( x ) = ln x , φ′( x ) = 1/ x . Thus the (indefinite) integral becomes

Indeterminate Forms 5.4 More on Improper Integrals

Thus

Indeterminate Forms 5.4 More on Improper Integrals

Thus the area under the curve and above the x -axis is 1/(3 ln 3 2).

Example 2

Because of inflation, the value of a dollar decreases as time goes on. Indeed, this decrease in the value of money is directly related to the continuous compounding of interest. For if one dollar today is invested at 6% continuously compounded interest for ten years then that dollar will have grown to e 0.06·10 = $1.82 (see Section 6.5 for more detail on this matter). This means that a dollar in the currency of ten years from now corresponds to only e −0.06·10 = $0.55 in today’s currency.

Now suppose that a trust is established in your name which pays 2 t + 50 dollars per year for every year in perpetuity, where t is time measured in years (here the present corresponds to time t = 0). Assume a constant interest rate of 6%, and that all interest is re-invested. What is the total value, in today’s dollars, of all the money that will ever be earned by your trust account?

Solution 2

Over a short time increment [ t j −1 , t j ], the value in today’s currency of the money earned is about

(2 t j + 50) · ( e −0.06· t j ) · Δ t j .

The corresponding sum over time increments is

Indeterminate Forms 5.4 More on Improper Integrals

This in turn is a Riemann sum for the integral

Indeterminate Forms 5.4 More on Improper Integrals

If we want to calculate the value in today’s dollars of all the money earned from now on, in perpetuity, this would be the value of the improper integral

Indeterminate Forms 5.4 More on Improper Integrals

This value is easily calculated to be $1388.89, rounded to the nearest cent.

You Try It : A trust is established in your name which pays t + 10 dollars per year for every year in perpetuity, where t is time measured in years (here the present corresponds to time t = 0). Assume a constant interest rate of 4%. What is the total value, in today’s dollars, of all the money that will ever be earned by your trust account?

Find practice problems and solutions for these concepts at: Indeterminate Forms Practice Test.

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