Question

积分0到1 √（1-x²） dx 等于多少

Answer 1

积分0到1 √（1-x²） dx
let
x = sina
dx = cosa da
x =0, a= 0
x=1, a =π/2
积分0到1 √（1-x²） dx
= 积分0到 π/2 (cosa)^2 da
= 积分0到 π/2 (cos2a+1)/2 da
=[ (sin2a)/4 + a/2] (0,π/2)
= π/4

Answer 2

作变换 x=sint 0<x<1所以t可以取值范围选(0<t<π/2)
而 dx = cost dt 根号(1-x^2) = |cost| = cost 因为0<t<π/2
这样积分0到1 根号(1-x^2)dx = 积分0到π/2 cost * cost dt 就变成 (cost)^2的积分了

Answer 3

几何上等于半径为1的圆面积的1/4，结果为π/4

Answer 4

结果是π/4 ， tllau38回答正确！

Answer 5

∫（0 1）√（1-x²）dx
=∫(0 π/2)cosxd(sinx)
=∫(0 π/2)cos²xdx
=1/2∫(0 π/2)[(cos(2x)+1]/2d(2x)
=(1/4)[sin(2x)+x]|(0 π/2)
=(1/4)(0+π/2-0-0)
=π/8