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AP Calc BC in a week - Part 3

Introduction

Today we'll be doing:

  • 6C Integrals involving Parametrically Defined Functions
  • 7C Arc Length
  • 7D Improper Integrals
  • 8B Motion along a Plane Curve

Let's start!

6C Integrals involving Parametrically Defined Functions

The book only gives us two examples. From these two examples I gather that we must substitute the parametric equations into \(x\) and \(y\), and convert the \(dx\) into \(d\) of whatever the parametric equations are in terms of.

Example: \(\int_0^4 xy dx\), \(x=4\sin\theta\), \(y=2\sin\theta\)

We would simply plug in \(x\) into \(x\) and \(y\) into \(y\). \(dx=4\cos\theta d\theta\), so we plug in that for \(dx\). The limits also have to be converted, which we can just do by plugging the limits into the \(x\) equations and finding \(\theta\) : \(0=\sin 0\), \(4=4\sin\frac{\pi}{2}\)

The integral then becomes \(\int_0^{\frac{\pi}{2}}4\sin\theta 2\sin\theta 4\cos\theta d\theta\) and we can compute it normally.

Take-home message

I don't think we need one.

  • Plug in \(x\) and \(y\) and the \(dx\) from the parametric equation, and convert the limits with the parametric equation.

7C Arc Length

Okay... hm. This looks like memorization.

So first thing to remember is this: $$ds^2 = dx^2 + dy^2$$

It kinda looks like Pythagoras. From the diagram in the book, it looks like \(ds\) is the arc, \(dx\) is the "run" and \(dy\) is the "rise". Kinda like a hypotenuse and two legs.

I suppose we can do something like

  • \(ds = \sqrt{dx^2 + dy^2}\)
  • \(s = \int_a^b \sqrt{dx^2 + dy^2}\)

So the find the arc length from parametric equations, we use: $$s = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2} dt$$ Where \(a\) and \(b\) are values of \(t\) where the arc is in between.

For the arc length of \(y=f(x)\), we use: $$s = \int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2} dx$$ Where \(a\) and \(b\) are values of \(x\) where the arc is in between.

And for the arc length of \(x=g(y)\), we use: $$s = \int_a^b \sqrt{1+\left(\frac{dx}{dy}\right)^2} dy$$ Where \(a\) and \(b\) are values of \(y\) where the arc is in between.

I have no trick for this. Just... memorize it. And practice it. You can try to use the pseudo-Pythagoras, I suppose. $$ds^2 = dx^2 + dy^2$$

7D Improper Integrals

There are two types of improper intergrals:

  1. At least one of the limits of the (definite) integral is infinite (making the interval unbounded)
  2. The function being (definitely) integrated has a point of discontinuity

I don't think I need to give examples.

Improper Integral 1 (infinite limit)

To compute this type of improper integral, we treat the infinite limit as a limit (haha). $$\int_a^\infty f(x) dx = \lim_{b\to\infty} \int_a^b f(x) dx$$

Then we just compute the definite integral, leaving \(b\) as the variable \(b\), and the apply the limit.

Example (from the book) \(\int_1^\infty \frac{1}{x^2} dx\):

  • \(= \lim_{b\to\infty} \int_1^b \frac{1}{x^2} dx\)
  • \(= \lim_{b\to\infty} -\frac{1}{x}|_1^b\)
  • \(= \lim_{b\to\infty} \left(-\frac{1}{b} - -\frac{1}{1} \right)\)
  • \(= \lim_{b\to\infty} \left(1 -\frac{1}{b} \right)\)
  • \(= 1 - \frac{1}{\infty}\)
  • \(=1\)

Apparently \(\int_1^\infty \frac{1}{x^n} dx\) converges if \(n>1\) and diverges if \(n\le 1\).

Uhh, practice with some examples to get the hang of it, I guess.

Improper Integral 2 (discontinuous function)

To compute this type of improper integral, we make the integral limits into limits. That's just another pun so I'll demonstrate:

If \(f(x)\) becomes infinite at \(a\), $$\int_a^b f(x) dx = \lim_{k\to a^+} \int_k^b f(x) dx$$

If \(f(x)\) becomes infinite at \(b\), $$\int_a^b f(x) dx = \lim_{k\to b^-} \int_a^k f(x) dx$$

If \(f(x)\) becomes infinite at at \(c\) and \(c\) is in between \(a\) and \(b\), $$\int_a^b f(x) dx = \lim_{k\to c^-} \int_a^k f(x) dx + \lim_{m\to c^+} \int_m^b f(x) dx$$

If the limits exist, then the integral converges and there is a solution. If the limits don't exist, then the integral diverges and there is no solution.

The construction of the limit-integrals should be intuitional. I'm not going to give an example for this one. The actual working is similar to the type 1 improper integral, just with the different limit set-up. Just do some practice problems.

The Comparison Test

Ah. This seems pretty easy. So this method is used to determine if an improper integral will converge or diverge by comparing it to an integral that we already know.

  1. Convergence: If we know that \(\int g(x) dx\) converges, and we know that \(f(x) \le g(x)\), then \(f(x)\) converges.
  2. Divergence: If we know that \(\int g(x) dx\) diverges, and we know that \(f(x) \ge g(x)\), then \(f(x)\) diverges.

This should be intuitional. In case 1, \(f(x)\) is always below \(g(x)\), and since \(g(x)\) converges, so must \(f(x)\). Case 2 is the opposite: \(f(x)\) is always above \(g(x)\), and since \(g(x)\) diverges, so must \(f(x)\). (non-rigorous proof)

I'm not going to give examples (since I'd just copy them from the book anyway) and just do some practice problems, as always.

Take-home message

  • Improper integrals have infinite bounds or points of discontinuity
  • To evaluate such integrals, turn the infinite bound(s) or point(s) of discontinuity into a limit
  • Comparison test: if a function is less than a converging function then it must also converge ; if a function is greater than a diverging function then it must also diverge

8B Motion along a Plane Curve

Mmm. Remember when we had parametric equations and position vectors and velocity vectors and acceleration vectors?

Apparently this section is basically problem-solving using those and calculus. Mostly stuff like "find the parametric equations of the motion of the <object>" or "find its position vector" or "find its <position/velcity/acceleration> at time t".

It shouldn't be that bad. Just know your stuff and practice some problems.

Conclusion

Sorry this post took so long and I'm starting to omit details. My style of studying/reviewing is getting the big picture and being able to piece together the smaller pictures from the big picture.

Next time will be:

  • 9C Euler's Method
  • 9E3 Logistic Growth

Ooh yay! Two more days! We're almost done~!


2016-03-23 Paul Elder

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