Unit 9: Improper Integrals
Stretching the interval to infinity, or letting the integrand blow up, and asking whether the area still adds up.
6 topics · 27 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Improper Integrals with an Infinite Limit
- How it worksHow does replacing infinity with a variable bound t and taking a limit turn an impossible integral ordinary?
- Worked exampleFind the integral from 1 to infinity of 1/x2 dx.
- Common mistakeWhy doesn't an integral with an infinite bound automatically diverge just because its interval is infinite?
- Worked exampleFind the integral from 0 to infinity of the exponential decay e−x dx.
- Going furtherWhat happens to the integral from 1 to infinity of 1/x2 dx if the lower limit is changed to 2?
Your turn1 of 2
Find the integral from 1 to infinity of 4/x4 dx.Improper Integrals with an Infinite Discontinuity
- How it worksHow does splitting the integral at the discontinuity and taking a one-sided limit handle the blow-up point?
- Common mistakeWhy doesn't a bounded interval like [0, 1] guarantee an integral over it is automatically finite?
- Worked exampleFind the value of the improper integral from 0 to 1 of 1/√x dx.
- Going furtherWhat happens when the discontinuity sits in the middle of the interval instead of at an endpoint?
Your turn1 of 2
Find the value of the improper integral from 0 to 3 of 4/√x dx.p-Integrals on [1, infinity)
- How it worksHow does integrating x−p and taking the limit as t goes to infinity pin the dividing line at p = 1?
- Worked exampleFind the integral from 1 to infinity of 1/x3 dx and state whether it converges.
- Common mistakeWhy doesn't p = 1 count as convergent when every exponent even slightly bigger does?
- Worked exampleFind the integral from 1 to infinity of 1/x dx and explain why it diverges.
Your turn1 of 2
Find the integral from 1 to infinity of 3/x4 dx and confirm it converges.The Comparison Test for Improper Integrals
- How it worksHow does sitting below a known convergent integral, or above a known divergent one, decide a harder integral's fate?
- Worked exampleShow me that the integral from 1 to infinity of 1/(x2 + 1) dx converges by comparing it to 1/x2.
- Worked exampleShow me that the integral from 1 to infinity of 1/√x dx diverges by comparing it to a p-integral.
- Common mistakeWhy doesn't a function being smaller than a divergent integral's function tell you anything about convergence?
- Going furtherWhat happens when the comparison inequality only holds for x past some starting point instead of for all x?
Your turn1 of 2
To show the integral from 1 to infinity of 1/(x2 + 5) dx converges, which comparison correctly proves it?Gabriel's Horn: Finite Volume, Infinite Surface
- How it worksHow does revolving y = 1/x for x from 1 to infinity around the x-axis generate a shape with convergent volume?
- Worked exampleFind the volume of Gabriel's horn, formed by revolving y = 1/x for x ≥ 1 around the x-axis.
- How it worksHow does the surface area integral for the same horn end up diverging even though the volume converges?
- Common mistakeWhy doesn't a region that looks like it stretches out to infinity have to enclose infinite area or volume?
- Going furtherWhat happens to the horn's volume and surface area when the curve is 1/x2 instead of 1/x?
Your turn1 of 2
Find the volume of the horn formed by revolving y = 4/x for x ≥ 1 around the x-axis.Improper Integrals with Two Problem Points
- How it worksHow do you pick a convenient interior point to break an integral into two pieces at once?
- Common mistakeWhy isn't it enough for just one of the two split pieces of an improper integral to converge?
- Worked exampleFind the integral from negative infinity to infinity of x times e−x^2 dx.
- Worked exampleFind the integral from 0 to infinity of 1/(x2 + 1) dx by splitting it at a convenient interior point.
Your turn1 of 2
Without computing any antiderivative, what is the integral from negative infinity to infinity of x e−x^2 dx, and why?