Question

∫∫e^(-y^2)dxdy,其中D是由x=0,y=x,y=2所围成的闭区域.

Answer 1

楼上算错,怎会算出负数呢!

D的顶点是：(0,0)、(0,2)、(2,2)

∫∫ e^(-y²) dxdy,Y型区域

= ∫(0~2) ∫(0~y) e^(-y²) dxdy

= ∫(0~2) ye^(-y²) dy

= (-1/2)∫(0~2) e^(-y²) d(-y²)

= (-1/2) • [e^(-y²)] |(0~2)

= (-1/2) • [e^(-4) - 1]

= (-1/2) • (1/e⁴ - 1)

= 1/2 - 1/(2e⁴) ≈ 0.49084