Improper Integrals

Integrating to infinity, or across a break.

12 problems · about 22 minutes · LIM-6.A

NameDate

Figure — use these to answer the problems

This curve runs to infinity at zero and flattens toward the axis far out. Both features can make an integral improper.

y = 1/x + 0

Point on the curve2

Figure 1

Straightforward practice. Get the method working first.

5 problems

p = 1.5 in the p-integral from 1 to infinity. Converge? 1 yes, 0 no.
Answer
2
Why must an interior asymptote be split at?
- Integrating through it produces a finite answer that is simply wrong.
- It is tidier to write that way.
p = 2 in the p-integral from 1 to infinity. Converge? 1 yes, 0 no.
Answer
p = 1. Converge? 1 yes, 0 no.
Answer
p = 0.5. Converge? 1 yes, 0 no.
Answer

The same ideas with more to keep track of.

3 problems

How many kinds of improper integral were named?
Answer
An asymptote sits inside the interval. Must the integral be split? 1 yes, 0 no.
Answer
The limit defining an improper integral is infinite. Converge? 1 yes, 0 no.
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each integral by whether it is improper.
Write each item under the heading it belongs to: An infinite upper limit · A vertical asymptote inside the interval · A polynomial on a closed interval · A continuous function on a finite interval
Improper
Proper
p = 4 in the p-integral from 1 to infinity. Converge? 1 yes, 0 no.
Answer
The Value
The integral of 1 ÷ x² from 1 to infinity. What is its value?
Answer
The p Test
The integral of 1 ÷ xᵖ from 1 to infinity with p = 3. Does it converge? 1 yes, 0 no.
Answer