Cogito
AP Calculus BC · Chapter 7 · Lesson 1
Improper Integrals
Integrating to infinity, or across a break.
12 problems · about 22 minutes · LIM-6.A
What this lesson teaches
The student evaluates improper integrals using limits and determines convergence.
- An integral is improper with an infinite limit or an infinite integrand.
- Replace the offending bound with a limit and evaluate.
- The p-integral converges for p > 1, mirroring the p-series.
Warm Up
Straightforward practice. Get the method working first.
5 problemsp = 1.5 in the p-integral from 1 to infinity. Converge? 1 yes, 0 no.
Answer 1
Why Yes.
Why must an interior asymptote be split at?
Answer Integrating through it produces a finite answer that is simply wrong.
Why The answer would be invalid.
p = 2 in the p-integral from 1 to infinity. Converge? 1 yes, 0 no.
Answer 1
Why Above 1.
p = 1. Converge? 1 yes, 0 no.
Answer 0
Why The boundary diverges.
p = 0.5. Converge? 1 yes, 0 no.
Answer 0
Why Below 1.
Build It Up
The same ideas with more to keep track of.
3 problemsHow many kinds of improper integral were named?
Answer 2
Why Infinite limit or infinite integrand.
An asymptote sits inside the interval. Must the integral be split? 1 yes, 0 no.
Answer 1
Why Otherwise the answer is wrong.
The limit defining an improper integral is infinite. Converge? 1 yes, 0 no.
Answer 0
Why Only a finite limit converges.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each integral by whether it is improper.
Answer Improper: An infinite upper limit, A vertical asymptote inside the interval · Proper: A polynomial on a closed interval, A continuous function on a finite interval
Why Either an infinite limit or an infinite integrand.
p = 4 in the p-integral from 1 to infinity. Converge? 1 yes, 0 no.
Answer 1
Why Above 1.
The Value: The integral of 1 ÷ x² from 1 to infinity. What is its value?
Answer 1
Why 1.
The p Test: The integral of 1 ÷ xᵖ from 1 to infinity with p = 3. Does it converge? 1 yes, 0 no.
Answer 1
Why Yes.