Today we started with a few warm-up problems to review material from Calc I.
Find the location of the maximum of \(f(x) = \dfrac{x}{2}+\dfrac{\sin x}{\sqrt{3}}\) on the interval \([0,\pi]\).
The function \(y = \sin x\) is a wave. How much area is under one arch of the sine wave?
To introduce this idea, we first looked at the area under one arch of the function \(\sin(2x)\). We graphed it on Desmos, and noticed that it is narrower than the arc of \(\sin x\) by a factor of 2. Likewise \(\sin(3x)\) is 3-times narrower. So the integrals should reflect this.
\[\int_0^{\pi/2} \sin(2x)\, dx = ? ~~~~~ \int_0^{\pi/3} \sin(3x) \, dx = ?.\]
Substitution gives us a systematic way to calculate integrals like this. The steps of integration by substitution are:
Step 1 Let \(u\) be the part of the function on the “inside”.
Step 2 Calculate the differential \(du\).
Step 3 Substitute using \(u\) and \(du\) to get an integral without any of the original variable.
We did all three of these examples in class.
\(\int \cos(2x) \, dx\)
\(\int (5x)^2 \, dx\) (Hint: you don’t have to use substitution for this one, but substitution will work too.)
\(\displaystyle \int \dfrac{2x}{(1+x^2)^3} \, dx\)
On Monday, we used several integration rules. Let’s do a quick review of the integration rules you should know:
Antidifferentives The integral is the opposite of the derivative. So if you have a function like \(\cos x\) that is the derivative of another function, then you know how to integrate it. Just undo the derivative.
Linearity Integration “plays nice” with addition, subtraction, and constant multiplication. You integrate term-by-term, and you can pull constants out of integrals.
Power Rule If \(x\) is a variable and \(p\) is a constant, then \[\int x^p \, dx = \tfrac{1}{p+1} x^{p+1}+C\] as long as \(p \ne -1\) (which wouldn’t make sense because you can’t divide by zero).
Those three rules plus the substitution technique are the only integration rules you will need to know for now.
This first example shows how the power rule works with linearity.
The second example is a reminder that if you recognize a derivative, then you know how to integrate it.
Finally we showed how to combine the ideas above with substitution:
It is important to know the difference between terms and factors.
Very important: Powers distribute to factors, but not to terms! So \((a^2b)^2 = a^4 b^2\) but \((x+y)^2 \ne x^2+y^2\).
\(\displaystyle \int_0^{\pi/4} \tan^4 x \sec^2 x \, dx\) (Hint: Remember that the derivative of \(\tan x\) is \(\sec^2 x\).)
\(\displaystyle \int_0^{\pi/2} \cos x \sin x \, dx\) (Hint: let \(u = \sin x\).)
\(\displaystyle \int_0^2 2x \sqrt{4-x^2} \, dx\)
Compare the integrals \(\int \sin (x^2) \, dx\) and \(\int \sin^2 (x) \, dx\). Both of these are difficult to compute, in fact, one is actually impossible. What additional factor would each integral need in order to make it amenable to the substitution technique?
Here is a challenge problem.
The trick is make a complete list of all the equivalences when you let \(u\) be the function inside the square root:
\[\begin{array}{cc} u = x^2 + 1 & du = 2x \, dx \\ x = \sqrt{u-1} & dx = \frac{du}{2x} = \frac{du}{2 \sqrt{u-1}} \end{array}\]
With these substitutions, the integral above becomes:
Not every integral needs a substitution. In each of the following, decide if substitution is helpful or not:
\(\displaystyle \int \frac{x+1}{x^3} \, dx\) We did this one in class, but didn’t have time for the other two.
\(\displaystyle \int \sqrt[3]{4+5x} \, dx\)
\(\displaystyle \int x^2(x^3+1) \, dx\)
Today we went over the problems in Homework 1.
Most of the problems involve a substitution. Here is a Khan Academy video that shows how to multiply by a constant if one is missing when you try a u-substitution: https://youtu.be/oqCfqIcbE10. This is a useful trick in a couple of the homework problems.
Some of the homework problems don’t need substitution. In Problem #6, you can distribute the power to both factors in \((4y)^{5/2}\). Then you get: \[\frac{1}{(4y)^{5/2}} = \frac{1}{4^{5/2} y^{5/2}}\] Then move the \(y\) to the top by making the power negative and you can do the integral. Also, remember that \((4)^{5/2} = (4^{1/2})^5 = 2^5 = 32\). So you just need to integrate \[\tfrac{1}{32} \int y^{-5/2} \, dy.\]
Another problem where a little algebra makes the problem easier is Problem #3. Notice that you can divide both terms in the numerator (top of the fraction) by the bottom to create two separate fractions. Then, \[\frac{1+\sqrt{x}}{\sqrt{x}} = \frac{1}{\sqrt{x}} + \frac{\sqrt{x}}{\sqrt{x}}\] Then, using the rules for subtracting powers in fractions, this can be simplified even more, to make an easy integral.
Definite integrals have these properties. These are easy to remember if you remember that \(\int_a^b f(x) \, dx\) represents the area under \(f(x)\) from \(x=a\) until \(x=b\).
\(\displaystyle \int_a^c f(x) \, dx= \int_a^b f(x) \, dx + \int_b^c f(x) \, dx\).
\(\displaystyle \int_a^a f(x) \, dx = 0\)
\(\displaystyle \int_b^a f(x) \, dx = - \int_a^b f(x) \, dx\)
If \(f(x)\) is odd, then \[\int_{-a}^a f(x) \, dx = 0.\] If \(g(x)\) is even, then \[\int_{-a}^a g(x) \, dx = 2 \int_0^a g(x) \, dx.\]
\(\displaystyle \int_{-\pi}^\pi x \cos (x^2) \, dx\)
\(\displaystyle \int_{-2}^{2} x^4 - 5x^2 + 3 \, dx\)
Unlike derivatives, it is not always possible to calculate integrals. Here are some simple functions that have no formula using standard functions for their antiderivatives:
\[\frac{\sin x}{x} ~~~~ \cos(x^2) ~~~~ \sin(\sqrt{x})\]
Question: What happens if you try to use the substitution \(u = x^2\) to integrate \(\displaystyle \int \cos(x^2)\) , dx? You might get \[\int \cos (u) \, dx,\] which looks simple, but you cannot integrate a function of \(u\) with respect to a different variable in the differential.
Even when you can’t integrate a function, you can still use the Riemann Sum formula to find the area under the curve:
\[\int_a^b f(x) \, dx = \lim_{n \rightarrow \infty} \sum_{k = 1}^n f(x_k) \Delta x\] where \(\Delta x = \frac{b-a}{n}\) and \(x_i = a + k \Delta x\).
The average value of a function \(f(x)\) on an interval \([a,b]\) is \[\frac{1}{b-a} \int_a^b f(x) \,dx\]
Rationale: average value of \(n\) distinct \(y\)-values would be the sum of the y-values divided by \(n\).
Find the average value of the points on the parabola \(y = 4 - x^2\) that are above the \(x\)-axis.
Find the average value of one arch of the \(\sin x\).
A rocket has a velocity (in meters per second) that is given by the formula \(v(t) = 10 t - 2t^2\) from time \(t = 0\) until \(t=5\). What is the average velocity of the rocket?
What is the average distance from a point on the interval \([0,3]\) to the point \(x=1\)?