課程名稱︰微積分 2 課程性質︰生物環境系統工程學系/工程科學及海洋工程學系/地質科學系必修 課程教師︰呂治鴻 開課學院：理學院 開課系所︰數學系 考試日期︰2025/12/08(一)，17:30-18:20 考試時限：50 分鐘 試題 : National Taiwan University - Calculus 2 (Class 09) - Quiz 2 2025/12/08 (Monday) - 50 minutes Name:＿＿＿＿＿＿＿＿＿＿＿＿ Student ID Number: ＿＿＿＿＿＿＿＿＿＿＿＿＿＿ There are FOUR questions in this quiz. Your work is graded on the quality of your writing as well as the validity of the mathematics. 1. (15 points) Find the following improper integrals. 1 2 (a) (5 points) ∫ x ln(x) dx. 0 ∞ exp(x) (b) (5 points) ∫ ─────── dx. -∞ 1 + exp(2x) 27 dx (c) (5 points) ∫ ──────. -8 x^(1/3) 2. (10 points) Determine whether the following improper integrals are convergent or divergent. ∞ (a) (5 points) ∫ exp(-x^4) dx. 1 ∞ ln(x) (b) (5 points) ∫ ───── dx. 1 x 3. (15 points) x (a) (5 points) Find the length of the curve y = ∫√(cos(2t))dt for 0 ≦ x≦4. 0 (b) (5 points) Consider the region bounded by the graphs of y = exp(x), y = 0, x = 0, and x = 1. Find the volume of the solid formed by revolving the region about the x-axis. (c) (5 points) Consider the region bonuded by the graphs of y = sin(x) and the x-axis for 0≦x≦π. Find the volume of the solid formed by revolving the region about the y-axis. 4. (10 points) Solve the following differential equations. (a) (5 points) y' + y tan(x) = (cos(x))^2. y^2 (b) (5 points) y' = ─────. 1 + x^2 --

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