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
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!!