Question

1.求不定积分x的平方cosxdx 2.求曲线y等于x的平方与y等于根号下x所围成图形面积

Answer 1

1. 积分过程如下：

2.  设面积为S，则

Answer 2

1
∫x^2cosxdx
=∫x^2dsinx
=x^2sinx-∫sinxdx^2
=x^2sinx-2∫sinx *xdx
=x^2sinx+2∫xdcosx
=x^2sinx+2xcosx-2∫cosxdx
=x^2sinx+2xcosx-2sinx+C

2
y=x^2与y=√x的交点是(0,0)及(1,1)
当0<x<1时
√x>x^2
所以
S=∫(0->1)(√x-x^2)dx
=∫(0->1)x^(1/2)dx-∫(0->1)x^2dx
=2/3 x^(3/2)|(0->1)-x^3/3 |(0->1)
=2/3-1/3
=1/3