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 \([0,\pi]\).
Find the indefinite integral \(\displaystyle \int 2x^3 + \frac{5}{x^2} \, dx\).
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.)
\(\displaystyle \int \dfrac{2x}{(1+x^2)^3} \, dx\)
Class was canceled today because of the weather, but I made a video where I solve several integration examples and explain my strategy for doing so. Hopefully it helps with the homework problems that are due on Monday:
If \(f(x)\) is odd, then \[\int_{-a}^a f(x) \, dx = 0.\] If \(g(x)\) is even, then \[\int_{-a}^a g(x) \, dx = 2 \int_0^a g(x) \, dx.\]
\(\displaystyle \int_{-2}^2 x^3 - 4x \, dx\)
\(\displaystyle \int_{-2}^{2} x^4 - 5x^2 + 3 \, dx\)
\(\displaystyle \int_{-\sqrt{\pi/2}}^{\sqrt{\pi/2}} x \cos (x^2) \, dx\)
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 \(x=1\)?