Question

数学微积分求面积体积问题

(1) 求由 y=x2与直线 y=2x 所围图形的面积；(2) 求该图形绕 x 轴旋转而成的旋转体的体积.

Answer 1

(1)
y=x^2
y=2x
x^2=2x
x=0 or 2
A
=∫(0->2) (2x-x^2) dx
=[ x^2 -(1/3)x^3]|(0->2)
=4- 8/3
=4/3
(2)
Vx
=π∫(0->2) [(2x)^2 -(x^2)^2 ] dx
=π[ (4/3)x^3 - (1/5)x^5]|(0->2)
=π ( 32/3 - 32/5)
=(64/15)π