- Your quiz today will be over integration, including techniques like substitution and integration by parts.
- James Hansen argues that global warming is accelerating, while the current administration erases information about climate change from the U.S. Government agencies charged with studying it (EPA, NASA, NOAA). It doesn't take an applied mathematician to tell you that warming is occurring:
(NASA video)
The question in James Hansen's mind is whether it's accelerating or not. That's a mathematical question, in many ways.
So what is our government doing in the face of it? USDA orders removal of climate change mentions from public websites (and this is happening to the EPA, and will happen elsewhere....).
- Last time we did a few more problems concerning Vol
2, Sec 3.1: Integration by Parts.
- Then we started Vol
2, Sec 3.2: Trigonometric Integrals. Let's pick back up
where we left off: looking at a few examples which illustrate
the importance of trigonometric identities. The examples we
looked at were
- Checkpoint 3.6: \[ \int \cos^{3}(x)\sin^{2}(x) d x \] and
- Checkpoint 3.7: \[ \int \cos^{2}(x) d x \]
- The trig form of the Pythagorean Theorem: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
- Sine of a sum: \[ \sin(\alpha+\beta)=\sin(\alpha)\cos(\beta)+\sin(\beta)\cos(\alpha) \]
- Cosine of a sum: \[ \cos(\alpha+\beta)=\cos(\alpha)\cos(\beta)-\sin(\beta)\sin(\alpha) \]
- Last time we observed that we can create the so-called "double-angle" formulas from these: \[ \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \] \[ \sin(2\theta) = 2\sin(\theta)\cos(\theta) \]
- Let's have a look at some parts of our handout for trig
integrals.
In particular, let's start at the identities section. Then you may choose a couple of the Questions to look at.
- One particularly useful set of functions that you'll encounter in
this section is the hyperbolic trig functions, which are defined
as follows:
\[
\sinh(x) \equiv \frac{e^x-e^{-x}}{2}
\]
\[
\cosh(x) \equiv \frac{e^x+e^{-x}}{2}
\]
\[
\tanh(x) \equiv \frac{\sinh(x)}{\cosh(x)}
\]
These functions behave in ways similar to the regular trig functions,
in spite of being based off of exponential functions. In particular,
- \(\sinh(x)\) is odd
- \(\cosh(x)\) is even.
- \(\sinh'(x) = \cosh(x)\)
- \(\cosh'(x)=\sinh(x)\)
And there are other familiar-looking identities (but with some slight changes in the symmetries, as you may have noticed above in the derivative for cosh):
\[ \begin{align} 1 &= \cosh^2(x)-\sinh^2(x) \\ \sinh(x + y) &= \sinh x \cosh y + \cosh x \sinh y \\ \cosh(x + y) &= \cosh x \cosh y + \sinh x \sinh y \\ \end{align} \]
- Let's have a closer look at a couple of these cases:
- $\displaystyle
\int^{}_{} \sqrt{a^2-x^2}\ d x$
- $\displaystyle \int^{}_{} \sqrt{1+x^2}\ d x$ (in this case, we'll make use of those hyperbolic trig functions to solve it).
- $\displaystyle
\int^{}_{} \sqrt{a^2-x^2}\ d x$
- Then check out our worksheet for this unit.
- Finally we'll have a quiz.
- Check out my Trigonometric Substitution summary (from a previous class).
- Check out my Trigonometric Integrals summary (from a previous class).
- Try out the integration animation which is mentioned in the text. I like Desmos, and certainly hope that you'll find it useful.
- Here's a L'Hopital worksheet for you to try (a bit of "review" of the inverse trig functions, too).
- Mathematica on-line is an option, if you are at a computer without Mathematica installed. I've tried it and it worked pretty well!