Problems

01

Comparison test

Use the comparison test to determine whether the integral converges:

$${\int }_1^{\infty }\frac{dx}{x^4+\sqrt{x}}$$

02

Proper vs. improper

For each integral below, determine whether it is proper or improper, and if improper, explain why.

(a) ${\displaystyle {\int }_0^2\frac{dx}{x^{1/3}}}$ $$ (b) ${\displaystyle {\int }_0^1{e}^{-x}dx}$ $$ (c) ${\displaystyle {\int }_0^{\pi /2}\sec xdx}$

(d) ${\displaystyle {\int }_0^{\infty }\sin xdx}$ $$ (e) ${\displaystyle {\int }_1^{\infty }\ln xdx}$ $$ (f) ${\displaystyle {\int }_0^3\ln xdx}$

03

Gabriel’s Horn - Volume and surface of revolution

The curve $y=\frac{1}{x}$ for $x>1$ is rotated about the $x$-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 $p$-integral which is divergent.

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) ${\displaystyle {\int }_1^{\infty }\frac{1}{x^{19/20}}dx}$ $$ (b) ${\displaystyle {\int }_0^5\frac{1}{x^{20/19}}dx}$ $$ (c) ${\displaystyle {\int }_1^{\infty }{e}^{-x}dx}$

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) ${\displaystyle {\int }_0^1\ln xdx}$ $$ (b) ${\displaystyle {\int }_{-\infty }^{+\infty }x{e}^{-x^2}dx}$ $$ (c) ${\displaystyle {\int }_1^2\frac{1}{{(x-1)}^2}dx}$