Question

设D为曲线y=1-x平方,直线y=x+1及x轴所围成的平面图形（1）求平面图形D的面积S

Answer 1

（1）面积S=∫[(1-y)^(1/2)-y+1]dy

=[(-2/3)(1-y)^(3/2)-y^2/2+y]│

=2/3-1/2+1

=7/6

（2）旋转体的体积=π∫(x+1)²dx+π∫(1-x²)²dx

=π[(x+1)³/3]│+π[x-2x³/3+x^5/5]│

=π(1-0)/3+π(1-2/3+1/5)

=13π/15.