Review of week eight and preview of week nine (homeowk in separate post)

Hi all,

Last week we covered trig integrals, trig substitutions and rational integrals. Next week, we will finish with rational integrals and introduce improper integrals. Just to remind you, a test is coming up, so make sure to review!

The major goal for trig substitution is to handle nonlinear functions such as fraction exponents, and typical examples we see are summed up in the table on page 467. By doing these substitutions and using the trig identities, we are left with one whole piece under the square root. Therefore, if you see strange functions under the square roots, try to complete the square and leave only constant outside– an example is Example 7 on page 471.  Start making your note sheet and make sure to write down the trig identities on your sheet for your test!

After your trig substitution, you should get an integral shaped like the ones in section 7.2: product of powers of sine and cosine, or product of powers of tangent and secant. Use the general methods we studies in section 7.2 and classify the resulting trig integrals by the powers of the trig functions. The results are summed up in tables on page 462 and 463.

Rational integrals have integrands that are rational functions. Any rational functions can always be simplified to the sum of a polynomial and a proper rational function. By proper, we mean the numerator has degree strictly lower than the degree of denominator. If the rational function is already proper, we do partial fractions and decompose  the integrand into things we know the antiderivative of; if not, first simplify by long division.

We covered the case where the results of partial fractions are distinct linear factors, next week we will study the cases where the results of the partial fraction contain repeated factors or irreducible quadratic factors.

The third true and false problem in our test 1 contains an integral that has a undefined spot in the integration domain, such integrals are one kind of improper integrals. Improper integrals are definite integrals that contain either undefined points in the integration interval, or has infinity as one integration limit. For these integrals, we need to be careful and check if it makes sense to compute them first.

Homework problems are in a separate post, make sure to check them out!

Alright, hope you have a good weekend, and don’t forget to do quiz corrections, they are due Nov 12/13!


Review of week seven and preview of week eight (Homework due Thursday 29th/Friday 30th)

Hi all,

Last week we finished with the two applications of integration: for definite integrals, we applied the idea of Riemann sum to compute the volumes of solids; for indefinite integrals, we studied the separable differential equations.

In computation of volumes, we use the same idea as computation of areas under curve: cut the big solid into small pieces that we can approximate with regular shapes, add up the small pieces and take the limit as the cut gets more and more fine. This process leads to a definite integral that we can compute. Areas (from chapter 5) were approximated by long thin strips of rectangles, and solids are approximated by thin slices or thin shells.

We mostly focus on solids of revolutions, as they are easy to compute. With slicing, the thickness of your slice determines the variable in your integral. Find outer radius and inner radius in that variable, and compute the area of cross section by \pi R^2-\pi r^2. The volume is (\pi R^2-\pi r^2) thickness. With shell method, the thickness of your shell determines the variable in your integral. Find the radius and height of your shell, and compute the volume by 2\pi radius height thickness. These quantities are normally related to the x or y coordinates of the points on the bounding curves, depends on the rotating axis.

Differential equations contains a large class of different types of equations, as long as they are equations with derivatives. We look at separable equations, the kind that you can separate independent variable and unknown function onto two sides of the equation. This way, on each side of the equation you see a differential form explicit in only the independent variable or the unknown function. To solve the differential equation, integrate both sides respectively. Remember to add the arbitrary constant C and check for zeros when dividing by the unknown function. If you are given an initial conditions, use it to determine the constant C.

Next week, we will talk about the last technique in integration: trig substitution. Trig functions come with a lot of identities that can help simplify integrands, therefore are good to try if u-sub or IBP don’t work. Please review the identities in our textbook for next week!

There is a quiz on Monday/Tuesday, on volumes (both set up and compute).

Finally, here is the homework:

Reading: 9.3,7.2,7.3

9.3: 2,3,5,9,10,12,13,16,19,35

7.2: 7,11,12,19,51(no need to graph)

7.3: Read 1-3, do 6,7,9,10,18

Alright, that is it for now. See you next week and have a good long weekend!

Review of week six and preview of week seven, Homework (due 21 Wednesday/22 Thursday)

Hi all,

Last week we covered two methods of computing the volume: the slicing method and the shell method. Next week we will finish talking about the two methods and introduce my favorite topic in math: differential equations!

The two methods of slicing and shell are mostly used in computing the volume of solids of revolution, but you can also use them to compute the volume of general solids (for example, exercises 51 to 71 in section 6.2). The difference between the two methods is the small piece of volume you use for adding up: in slicing, small pieces of volume are thin slices with a base area (like sliced bread); in shell, small pieces of volume are thin shells with height and radius (like one layer in the bottom of a chuck of leek or green onion). Therefore in slicing, focus on finding the area of the cross section; and in shell method, focus on finding the height and the radius. To find out which variable is used in your integral, look at the thickness of one slice in slicing method, and the thickness of the shell in shell method.

Next week, we will see more exercises on computing volumes, and you will choose whether to choose slicing or shell method. The principle is always use whichever is easier to set up and compute.

The differential equations we will see are called separable equations. The idea is basically to translate your problems into two separate integrals. You will see then why it is important to add the arbitrary constant for indefinite integrals: the constants are related to the initial conditions.

There is a quiz on Thursday/the Monday of 26th. I will put up a practice quiz sometime next week.

Finally, here is the homework:

6.2: 25-30, 33,34,41,43,63

6.3: 1,2,3,7,8,11,23,24,30,38,40

Make sure to do quiz/test correction, the deadline is next Friday!!

Alright, have a good weekend and let me know if you have any questions!!

Review of week five and preview of week six

Hi all,

Last week, we had our first test on Tuesday/Wednesday, and we started talking about the application of definite integral. Remember, you can write up the corrections of your mistake from the test and get up to half of the points back and you can come to me for help, even with writing up of the corrections!

We have talked about the first application of definite integrals: to compute area between curves. Same idea as definite integral, we look at the thin long rectangles that approximate the thin long strips, add up all the approximate areas and take the limit. Since we are looking at areas, our heights of rectangles have to be non-negative, and you want to integrate the absolute values of the differences of the functions. Depending on the curves, choose the appropriate direction to cut up your areas!

Next week, we will spend most of our time writing out Riemann sums. The idea of the process is to find expressions of little pieces of element, add them up and take the limit. After the limiting process, you will obtain a definite integral, and we can solve the definite integral with any of the methods we have learned. Next week is going to be very geometric, so get ready to draw pictures!

I will post your homework over the weekend and it will be due next Thursday.

Alright, that is it for all, have a good weekend!

Review of week four and preview of week five

Hi all,

Last week we finished with substitution and integration by parts. A practice test was handed out to help you prepare the test coming up next Tuesday/Wednesday. We will then start Chapter 6: Applications of integration.

Both substitution and integration by parts are techniques for finding antiderivatives, so the goal for both is to simplify the integrand so it fits into the table on page 392. To differentiate the two methods: substitution normally involves functions that look “non-basic”, i.e., functions that are composed of different functions; integration by parts normally involves product of two functions. Since you rarely see only substitution or IBP, it is a good idea to try first do a substitution and simplify your integrand. Then going from there, it might be clear to you whether to do more substitution or IBP. For both methods, remember to follow the general tips/orders carefully and patiently, you should be fine!

One thing I noticed: when you have decided on your u (both in substitution and IBP) and look for du, remember du means the “differential of u”, therefore you want to differentiate u!!. Your goal is to simplify the integrand, so if it looks like it is getting more complicated, check your du!

Next Thursday/Friday, we will start with applications of integrations, in particular, the definite integration. The key idea behind this chapter is the same idea we had for Riemann sum: cut things into little pieces and add them up one by one, take the limit as your cut as more and more fine and denote the limit as a definite integral. Once you have obtained a definite integral, you can go back to the three techniques we learned to compute the definite integral (see sum up of quiz 1). It is a very visual and fun chapter!

There is no written assignment due this week, but you are probably busy reviewing for the test (everything we have learned so far). If you want me to check your work, just take a picture of it and email it to me. The more work you show, the easier it will be for me to figure out the mistake/good idea you had.

Finally, remember to get some free points back by regrading!!

Alright, see you soon!