求∫∫y/x^6dxdy,D是y=x^4-x^3的上凸弧段部分与x轴所形成的曲边梯形.

Question

求∫∫y/x^6 dxdy,D是y=x^4-x^3的上凸弧段部分与x轴所形成的曲边梯形.

Answer 1

∫∫_D f(x,y) dxdy
= ∫(0→1) dx ∫(0→x⁴ - x³) y/x⁶ dy
= (1/2)∫(0→1) (1/x⁶)[y²]：(0→x⁴ - x³) dx
= (1/2)∫(0→1) (1/x⁶)(x⁴ - x³)² dx
= (1/2)∫(0→1) (x⁸ - 2x⁷ + x⁶)/x⁶ dx
= (1/2)∫(0→1) (x - 1)² d(x - 1)
= (1/2) · [(x - 1)³/3]：(0→1)
= (1/6)[(1 - 1)³ - (0 - 1)³]
= 1/6