Question

Problem Zero: Read the section and make your own summary of the material.

Short answer

The summary of the material contains,

• The condition for the Improper Integrals over Unbounded Intervals.
• The condition for the Improper Integrals over Discontinuities.
• The condition for the improper Integrals of Power Functions on $[1,\infty )$.
• The condition for the improper Integrals of Power Functions on $\left[0,1\right]$.
• Comparison Test for Improper Integrals.

Step-by-step solution

Step 1. Given information

We need to read the section and have to make our own summary of the material.

Step 2. The summary of the material is,

• The condition for the improper integrals over the discontinuities: Suppose $a$and $b$are any real numbers and $f\left(x\right)$is a function,

1. If $f$is continuous on $(a,b]$but not at $x=a$, then ${\int }_a^{b}f\left(x\right)dx=\underset{A\to a^{+}}{\lim }{\int }_A^{b}f\left(x\right)dx$.
2. If $f$is continuous on $[a,b)$but not at $x=b$, then localid="1649402595952" ${\int }_a^{b}f\left(x\right)dx=\underset{B\to b^{-}}{\lim }{\int }_a^{B}f\left(x\right)dx$.
3. If localid="1649402605060" $f$is continuous on localid="1649402600545" $[a,c)\cup (c,b]$but not at localid="1649402614665" $x=c$, then we can define the integral over localid="1649402609560" $\left[a,b\right]$as the sum of the first two cases:

localid="1649402621438" ${\int }_a^{b}f\left(x\right)dx=\underset{B\to c^{-}}{\lim }{\int }_a^{B}f\left(x\right)dx+\underset{A\to c^{+}}{\lim }{\int }_A^{b}f\left(x\right)dx$.

• The condition for the improper integrals over the unbounded intervals:

1. If localid="1649402645957" $f$is continuous on localid="1649402652701" $[a,\infty )$, then localid="1649402629247" ${\int }_a^{\infty }f\left(x\right)dx=\underset{B\to \infty }{\lim }{\int }_a^{B}f\left(x\right)dx$.
2. If localid="1649402666806" $f$is continuous on localid="1649402659678" $(-\infty ,b]$, then localid="1649402673804" ${\int }_{-\infty }^{b}f\left(x\right)dx=\underset{A\to -\infty }{\lim }{\int }_A^{b}f\left(x\right)dx$.
3. If localid="1649402688295" $f$is continuous on localid="1649402680730" $\left(-\infty ,\infty \right)$, then define the integral over localid="1649402694869" $\left(-\infty ,\infty \right)$as a sum of the first two cases, splitting at any real number localid="1649402638130" $c$.

Step 3. Let us see the condition for the power functions.

• The condition for the improper integrals of the power function on $[1,\infty )$:

1. If $0<p\le 1$, then $\frac{1}{x^p}\ge \frac{1}{x}$for $x\in [1,\infty )$and ${\int }_1^{\infty }\frac{1}{x^p}dx$diverges.
2. If $p>1$, then $\frac{1}{x^p}<\frac{1}{x}$for $x\in [1,\infty )$and ${\int }_1^{\infty }\frac{1}{x^p}dx$converges to $\frac{1}{p-1}$.

• The condition for the improper Integrals of Power Functions on $\left[0,1\right]$:

1. If $0<p<1$, then $\frac{1}{x^p}<\frac{1}{x}$for $x\in \left[0,1\right]$and ${\int }_0^1\frac{1}{x^p}dx$converges to $\frac{1}{1-p}$.
2. If $p\ge 1$, then$\frac{1}{x^p}\ge \frac{1}{x}$for$x\in \left[0,1\right]$and${\int }_0^1\frac{1}{x^p}dx$diverges.

Step 4. Comparison test.

• The condition for the Comparison Test for Improper Integrals:

Suppose $f$and $g$are functions that are continuous on an interval $I$.

1. If the improper integral of $f$on $I$converges and $0\le g\left(x\right)\le f\left(x\right)$for all $x\in I$where the functions are defined, then the improper integral of $g$on $I$also converges.
2. If the improper integral of $f$on $I$diverges and $0\le f\left(x\right)\le g\left(x\right)$for all $x\in I$where the functions are defined, then the improper integral of $g$on $I$also diverges.