The shell method uses the formula 2pi int x*f(x) color(white)(.) dx2π∫x⋅f(x).dx with f(x)f(x) being our function and xx as our radius
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
y=xy=x
2pi int x * x color(white)(.)dx2π∫x⋅x.dx
2pi int x^2 color(white)(.) dx2π∫x2.dx
2pi xx x^(2+1)/(2+1)2π×x2+12+1 via power rule
2pi xx x^3/32π×x33
(2 pi x^3)/32πx33
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
y = -x/2y=−x2
2pi intx * -x/2 color(white)(.)dx2π∫x⋅−x2.dx
(-2pi)/2 int x^2color(white)(.) dx−2π2∫x2.dx
-pi intx^2color(white)(.)dx−π∫x2.dx
(-pi x^3)/3−πx33
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
y = x^2y=x2
2pi int x * x^2 color(white)(.) dx2π∫x⋅x2.dx
2pi int x^3 color(white)(.) dx2π∫x3.dx
2pi xx x^4/42π×x44
(pix^4)/2πx42
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
y = 2-xy=2−x
2pi int x(2-x) color(white)(.) dx2π∫x(2−x).dx
2pi int 2x - x^2 color(white)(.) dx2π∫2x−x2.dx
2pi int2x color(white)(.) dx - 2pi int x^2 color(white)(.) dx2π∫2x.dx−2π∫x2.dx
4pi intx color(white)(.) dx - (2pi x^3)/34π∫x.dx−2πx33
4pi x^2/2 - (2pix^3)/34πx22−2πx33
2pi x^2 - (2pix^3)/32πx2−2πx33
(3(2pix^2))/3 - (2pix^3)/33(2πx2)3−2πx33
(6pix^2 - 2pix^3)/36πx2−2πx33
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
