Calculus Volume 2 · free preview
Improper Integrals
Integrals over infinite intervals and across vertical asymptotes are defined as limits, converging when the limit exists and diverging otherwise, with the p-test and comparison test deciding convergence.
1. Infinite intervals
Infinite intervals
2. Unbounded integrands
Unbounded integrands
3. Comparison and limit comparison
Comparison and limit comparison
4. The gamma function preview
The gamma function preview
5. Definition at infinity
How is the improper integral from a to infinity of f(x) dx defined?
6. P-test threshold
The integral from 1 to infinity of dx/x^p converges for which exponents?
7. Singularity p-test reversal
The integral from 0 to 1 of dx/x^p across the singularity at zero converges for which exponents?
8. Limit comparison condition
The limit comparison test for integrals from 1 to infinity requires which condition to relate f and g?
9. Doubly infinite splitting
To evaluate the integral from minus infinity to infinity of f(x) dx, where both tails are improper, what does the definition require?
10. One over x squared at infinity
The integral from 1 to infinity of dx/x^2 converges to what number? Answer with a number.
11. Log at one divergence
The integral from 0 to 1 of dx/x equals the limit of ln t as t approaches 0 from the right, with sign. This improper integral does what? Answer with one word, converges or diverges.
12. Gamma at one
The gamma function Gamma(1) equals the integral from 0 to infinity of e^(-x) dx. What number is it? Answer with a number.
13. Gamma recursion
Integration by parts on the gamma function Gamma(n + 1) equals integral from 0 to infinity of x^n e^(-x) dx gives which recursion?
14. Gabriel horn volume
Gabriel's horn volume is pi times the integral from 1 to infinity of dx/x^2. As a number, what is it? Answer with a number.
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