Question

一道定积分求旋转体体积题

题目如图，这个体积咋求，大佬写出详细步骤！！

Answer 1

y=(1/2)x^2 (1)
y=x+4 (2)
sub (2) into (1)
x+4= (1/2)x^2
x^2-2x-8=0
(x-4)(x+2)=0
x=-2 or 4
Vx
=π∫(-2->4) [( y2)^2 -(y1)^2 ] dx
=π∫(-2->4) { (x+4)^2 -[(1/2)x^2]^2 } dx
=π∫(-2->4) [(x+4)^2 -(1/4)x^4 ] dx
=π [ (1/3)(x+4)^3 - (1/20)x^5]|(-2->4)
=π [ ( 512/3 - 1024/20) -( 8/3 +32/20) ]
=π [ 504/3 - 264/5 ]
=(576/5)π