01
Comparison test
Use the comparison test to determine whether the integral converges:
Solution
03
(1) Find comparable integrand:
Higher power dominates for large
: Therefore, compare to
.
(2) Make comparison:
And:
because it is a
-integral with . By the Comparison Test, we conclude that:
Link to original
02
Proper vs. improper
For each integral below, determine whether it is proper or improper, and if improper, explain why.
(a)
(b) (c) (d)
(e) (f)
Solution
06
(a) Improper: integrand
as . Note: this converges too, since it’s a
-integral to zero with . (b) Proper: no source of infinity.
Note: automatically converges.
(c) Improper:
as . Note: this diverges. Antiderivative is
as . (d) Improper: infinite upper bound.
Note: this diverges. Antiderivative is
which has no limit as . (e) Improper: infinite upper bound.
Note: this diverges. Antiderivative is
as . (L’Hopital’s rule, indeterminate form.) (f) Improper: infinite integrand at
. Note: this converges. Antiderivative is
Link to originalas . (Same indeterminate form as (e).)
03
Gabriel’s Horn - Volume and surface of revolution
The curve
for is rotated about the -axis. The resulting shape is Gabriel’s Horn. (a) Find the volume enclosed by the horn by evaluating a convergent improper integral.
(b) Show that the surface area of the horn is infinite by applying comparison to a
-integral which is divergent.
Solution
07
Volume:
Surface area:
But notice this:
But
diverges! So by the comparison test,
Link to originaldiverges as well.
04
Computing improper integrals
For each integral below, give the limit interpretation of improper integral and then compute the limit. Based on that result, state whether the integral converges. If it converges, what is its value?
(a)
(b) (c)
Solution
09
(a)
(1) Definition of improper integral:
(2) Antiderivative and limit:
(b)
(1) Definition of improper integral:
(2) Antiderivative and limit:
(c)
(1) Definition of improper integral:
(2) Antiderivative and limit:
Link to original
05
Computing improper integrals
For each integral below, give the limit interpretation of improper integral and then compute the limit. Based on that result, state whether the integral converges. If it converges, what is its value?
(a)
(b) (c)
Solution
10
(a)
(1) Definition of improper integral:
(2) Antiderivative and limit:
Note A: Use L’Hopital:
(b)
(1) Definition of improper integral:
(2) Antiderivative and limit:
(c)
(1) Definition of improper integral:
(2) Antiderivative and limit:
Link to original