Đề Ca2
𝐶â𝑢 1
𝑥2+ 𝑦2+ (𝑧 − 1)2= 1
𝑛⃗ = ±(𝑥, 𝑦, 𝑧 − 1)
𝑉𝑇𝑃𝑇 đơ𝑛 𝑣ị 𝑡ạ𝑖 𝑀 (1
2;
1
√2;
1
2) 𝑡ℎ𝑢ộ𝑐 𝑛ữ𝑎 𝑑ướ𝑖 𝑐ủ𝑎 𝑚ặ𝑡 𝑐ầ𝑢 , 𝑉𝑇𝑃𝑇 ℎướ𝑛𝑔 𝑣à𝑜 𝑡𝑟𝑜𝑛𝑔 𝑛ê𝑛 𝑙à 𝑛⃗
= (−1
2 ;
−1
√2;
1
2)
𝐶â𝑢 2
∬ 𝑥𝑦𝑑𝑥𝑑𝑦 = ∫ 𝑑𝑥
0
−2
∫ 𝑥𝑦𝑑𝑦 + 2−𝑥
−2𝑥
∫ 𝑑𝑥 1
0
∫ 𝑥𝑦𝑑𝑦 =−10
3 + 2−𝑥
√2𝑥−𝑥 2
1 4
𝐶â𝑢 3
𝐺ọ𝑖 𝐶1 𝑙à đườ𝑛𝑔 𝑡ℎẳ𝑛𝑔 𝑦 = 0 𝑐ó 𝑐ℎ𝑖ề𝑢 đ𝑖 𝑡ừ 𝑥 = 0 đế𝑛 𝑥 = −2
Á𝑝 𝑑ụ𝑛𝑔 𝐺𝑟𝑒𝑒𝑛 𝑡𝑎 đượ𝑐 𝐼𝐺= ∬ −2𝑦𝑑𝑥𝑑𝑦 = ∫ 𝑑𝜑
𝜋
0
∫ −2𝑟𝑠𝑖𝑛𝜑 𝑟𝑑𝑟 = −4
3 1
0
𝐼1= ∫ 𝑒𝑥𝑑𝑥 = −1 + 𝑒−2
−2
0
𝑆𝑢𝑦 𝑟𝑎 𝐼 = 𝐼𝐺− 𝐼1=−1
3 − 𝑒
−2
𝐶â𝑢 4 (𝑐ó 𝑛ℎ𝑖ề𝑢 𝑐á𝑐ℎ 𝑛ℎư𝑛𝑔 𝐺𝑎𝑢𝑠𝑠 𝑙à 𝑘ℎỏ𝑒 𝑛ℎấ𝑡)
𝐺ọ𝑖 𝑆1 𝑙à 𝑚ặ𝑡 𝑧 = 0 𝑐ó 𝑉𝑇𝑃𝑇 𝑛⃗⃗⃗⃗ = (0,0, −1) ℎướ𝑛𝑔 𝑥𝑢ố𝑛𝑔 1
𝑆2 𝑙à 𝑚ặ𝑡 𝑧 = 1 𝑐ó 𝑉𝑇𝑃𝑇 𝑛⃗⃗⃗⃗ = (0,0,1) ℎướ𝑛𝑔 𝑙ê𝑛 1
Á𝑝 𝑑ụ𝑛𝑔 𝐺𝑎𝑢𝑠𝑠 𝑡𝑎 đượ𝑐 𝐼𝐺 = ∭ 3𝑑𝑥𝑑𝑦𝑑𝑧 = 3𝑉 = 3𝜋
𝐼1= ∬ −𝑑𝑆 = −𝜋 ( 𝑠ử 𝑑ụ𝑛𝑔 𝑐á𝑐ℎ (𝑃, 𝑄, 𝑅) 𝑛⃗⃗⃗⃗ ) 1
𝐼2= ∬ 2𝑑𝑆 = 2𝜋
𝐼 = 𝐼𝐺− 𝐼1− 𝐼2= 2𝜋
Trang 2𝐶â𝑢 5
𝐶â𝑢 6
𝐶â𝑢 7