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:
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!