find the volume of the solid generated by revolving the shaded region about the given axis. x-axis. from 0 to 2

Question

y =-2 x+4

Answer 1

Volume of the solid, generated by the revolving  around x-axis is  found by

 S=π∫[f(x)]²dx, Integration bounds 0 and 2

S=π∫(2x+4)²dx=4π∫(x+2)²dx =4π(x+2)³/3|₀ ²=(4π/3)(5³-2³)=..

=(4π/3)(7)(19).

Since y=2x+4 is linear function, therefore when revolving around x-axis the solid body would be a cone. Height of the cone h equals to 2 and the radius of the basic is 8 (2*2+4). So you can find a volume of the cone with those parameters as well: h=2, r=8.