∫dy∫y^(1/3)cosx^5dx

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∫dy∫y^(1/3)cosx^5dx
∫dy∫y^(1/3)cosx^5dx

∫dy∫y^(1/3)cosx^5dx
积分域为x=y^(1/3),x=1,y=0围成的区域
即y=x³,x=1,y=0围成的区域
交换积分次序
∫(0,1)dy∫(y^(1/3),1) y^(1/3)cos(x^5) dx
=∫(0,1)dx∫(0,x³) y^(1/3)cos(x^5) dy
=∫(0,1) cos(x^5)*(3/4) [(x³)^(4/3)-0] dx
=(3/4)∫(0,1) cos(x^5)* x⁴ dx
=(3/4)*(1/5)sin(x^5)|(0,1)
=(3sin1)/20