Class 17 - Improper integrals (10/12/2017, Thursday)
2nd Gateway
You must pass the second Gateway by Oct 13, Friday. Otherwise, your grade will be damaged a lot!
Webwork
- 8.3 due 10/18 Wed
- 8.3 due 10/18 Wed
Review: Polar coordinates
Today: Imporper integrals
We start by an example. You know how to compute
\int_{0.001}^{2}\frac{1}{x}dx = \ln(2) - \ln(0.001).
So what if we replace 0.001 by 0? Then you will have to consider
\int_{0}^{2}\frac{1}{x}dx,
but you would immediately say "You cannot talk about 1/0 in Math 116." You are right, but as a notation, we define
\int_{0}^{2}\frac{1}{x}dx := \lim_{\epsilon \rightarrow 0+}\int_{\epsilon}^{2}\frac{1}{x}dx.
How do we compute this? For any \epsilon > 0, we can compute
\int_{\epsilon}^{2}\frac{1}{x}dx = \ln(2) - \ln(\epsilon),
so
\lim_{\epsilon \rightarrow 0+}\int_{\epsilon}^{2}\frac{1}{x}dx = \lim_{\epsilon \rightarrow 0+}(\ln(2) - \ln(\epsilon)) = \infty,
or in other words, the limit diverges to the \infty, so the limit does not exists.
In fact, drawing a neat picture of rectangles can tell you that
\int_{0}^{\infty}\frac{1}{x}dx \geq 1 + 1/2 + 1/3 + 1/4 + \cdots.
However, we also can see that
\begin{align*}1 + 1/2 + 1/3 + 1/4 + \cdots &\geq 1 + 1/2 + 1/4 + 1/4 + 1/4 + 1/4 + \cdots \\ &= 1 + 1/2 + 1/2 + 1/2 + \cdots = \infty,\end{align*}
so we get
\int_{0}^{\infty}\frac{1}{x}dx = \infty,
in a geometric fashion.
Exercise. Compute
\int_{1}^{\infty}\frac{1}{x}dx = \lim_{r \rightarrow \infty}\int_{1}^{r}\frac{1}{x}dx.
(Hint: You can compute directly, or draw picture to realize that this integral is the same as the previous integral.)
Unlike the above exercise, we have
\begin{align*}\int_{1}^{\infty}\frac{1}{x^{2}}dx &= \lim_{r\rightarrow\infty}\left(\frac{-1}{r} + 1\right) \\ &= 1.\end{align*}
The integrals that require us to take limits are called improper integrals. We say an improper integral converges/diverges when the associated limit converges/diverges.
Exercise (p-test I). Show that when p > 1, the following integral converges
\int_{1}^{\infty}\frac{1}{x^{p}} dx.
Exercise (p-test II). Show that when p \leq 1, the following integral diverges
\int_{1}^{\infty}\frac{1}{x^{p}} dx.
Exercise (p-test III). Show that when p \geq 1, the following integral diverges
\int_{1}^{\infty}\frac{1}{x^{p}} dx.
(Hint: Use substitution u = 1/x and p-test II.)
Exercise (p-test IV). Show that when p < 1, the following integral converges
\int_{1}^{\infty}\frac{1}{x^{p}} dx.
(Hint: Use substitution u = 1/x and p-test I.)
Exercise. Compute
\int_{1}^{\infty}e^{-4x} dx.
Exercise. Find all real numbers c such that
\int_{1}^{\infty}e^{cx} dx
is convergent.
Warning. Your book (p.390) interprets the following integral to be divergent:
\int_{-1}^{1}\frac{1}{x} dx.
Exercise. Using p-test, determine whether the following improper integral is convergent:
\int_{0}^{\infty}\frac{1}{x^{2}} dx.
is convergent.
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