Tiêu đề 75 Multiple Integration.pdf ✅

MSTC 75: Multiple Integration In the case of board exams, the direct integration technique is the most appropriate to study in multiple integration. It features the evaluation of inner integrals and works outwards until the result. 1. Double Integration - Example:∫ ∫ (x 2 + y 2 )dy dx x 0 1 0 [SOLUTION] Solve the inner integral ∫ (x 2 + y 2 ) dy x 0 = 4 3 x 3 Therefore, ∫ ∫ (x 2 + y 2 )dy dx x 0 1 0 = ∫ 4 3 x 3dx 1 0 = 1 3 2. Triple Integration - Example: ∫ ∫ ∫ (x + y + z)dz dy dx 1−x−y 0 1−x 0 1 0 [SOLUTION] Integrate the innermost integral ∫ (x + y + z)dz 1−x−y 0 = (x + y)(1 − x − y) + 1 2 (1 − x − y) 2 = 1 2 (1 − x 2 − 2xy − y 2 ) Therefore, ∫ ∫ ∫ (x + y + z)dz dy dx 1−x−y 0 1−x 0 1 0 = ∫ ∫ 1 2 (1 − x 2 − 2xy − y 2 )dy 1−x 0 dx 1 0 Integrate further ∫ 1 2 (1 − x 2 − 2xy − y 2 )dy 1−x 0 = (1 − x) − x 2 (1 − x) − x(1 − x) 2 − 1 3 (1 − x) 3 = 1 6 (1 − x)(2 − x − x 2 ) Thus, ∫ ∫ 1 2 (1 − x 2 − 2xy − y 2 )dy 1−x 0 dx 1 0 = ∫ 1 6 (1 − x)(2 − x − x 2 )dx 1 0 = 1 8