【京大2009】カージオイドの長さ｜大学入試 数学 過去問 極方程式 積分

⭐️ "Oshie Math" allows you to ask Hayashi unlimited questions for a fixed monthly fee. You can try asking questions for free once! https://oshiemath.com/ ⭐️ "Hayashi Math Class," a specialized mathematics school where you can receive direct instruction from Hayashi. Please come for a trial lesson and interview! https://hayashi-math.com/ ✅ Official LINE for students applying to top universities: https://lin.ee/lI7n1SJ Subscriber benefits & live streams for applicants ℹ️ Shunsuke Hayashi's profile https://hayashishunsuke.com/profile/ ・Graduated from Sakae Higashi Junior High School → Chikuma High School → University of Tokyo, Faculty of Science, Department of Physics ・Scored 90% on the second-stage mathematics exam at the University of Tokyo, passing the exam as a current student ・2014 Japan Physics Olympiad Gold Medal ・Placed first place in the 2014 University of Tokyo Physics Mock Exam ℹ️ Please note ・The explanations are Shunsuke Hayashi's own and are not official university information. ・Amazon Associates links will be used when introducing books, etc. This problem, from the 2009 Kyoto University Science Mathematics A exam [6], asks you to find the length of a curve expressed as a polar equation. It's a famous curve called a cardioid! (However, since this problem is simply about finding the length of a curve, we won't focus on the name or shape, but will instead explain how to calculate it. If you're interested, try searching for "cardioid" or something similar.) The calculation is easy using the formula for the length of a curve expressed as a polar equation (introduced in the previous video)! Even without using that formula, you can derive the same integral form by converting it to x- and y-coordinates and calculating. ---------- Table of Contents 00:00 2009 Kyoto University Mathematics Course [6] 00:32 Explanation 1: How to think in xy coordinates (tedious) 01:13 Explanation 2: Calculating dx/dθ and dy/dθ 02:23 Explanation 3: Calculating dL/dθ from dx/dθ and dy/dθ 04:27 Explanation 4: Integrating dL/dθ to find the curve length 06:41 Explanation 5: How to use the formula for calculating curve length in polar coordinates (super convenient) 08:40 Summary: For polar equations, the formula in polar coordinates is overwhelmingly more convenient 09:30 Supplement: The curve for r = 1 + cosθ has a name 09:50 Conclusion