Today, we reviewed our list of applications of integrals and worked on 3 special examples. For each problem, the process of translating a sketch and problem description into integrals is EXACTLY the same as before.
However, the integrals we got cannot be solved by table or the substitution rule. We then discover how reversing the product rule for differentiation (like what we did on the chain rule for differentiation, to obtain substitution rule for integration) could help, and it is called integration by parts.
We fought with seemingly innocent examples in detail. However, as I had been sick, I made a few significant errors, some of which were not detected by you. Please check out the file now: wk4day8_wp. The most important mistake I did was on the observer’s picture for Q1, the axis on the cross section should be x-axis (instead of y-axis). Moreover, the y-axis being shrunk into a point indicates that as the cross-section moves in (or out) of the paper because it’s changing with y, not x.
The last example was about a trigonometric integral (Sec 7.3), which in my opinion is one of the most important integrals, and we will cover trigonometric integrals in details on Thursday. The application of these integrals will be found immediately after this, paving the way to integrals that require trigonometric substitutions (Sec 7.4).
EDITED: There is still one mistake in the hard integral \( \int \left( \arcsin y \right) dy \). I did not correct an extra factor of 1/2 in class, and now I typed the correct solution out and attach here for your reference: wk4day8_hard_int.