Integrating surfaces and solids of revolution with ���wheels���
P(p) is the parameter range of P.

P_(M*I*N)=-2,P_(M*A*X)=2.5

Sliders to set roots of polynomial f(p):

a=slider([-4,4,32])

b=slider([-4,4,32])

c=slider([-4,4,32])

d=slider([-4,4,32])

f=slider([-4,4,32])

Scaling factor for polynomial f(p):

S=slider([-1,1,100])

f(p) is the function to be revolved around axis.

function(f,p)=[if(S*[p-a]*[p-b]*[p-c]*[p-d]*[p-f],P_(M*I*N)<p<P_(M*A*X))]

function(P,p)=P_(M*I*N)*[1-p]+P_(M*A*X)*[p]

Area of sector (of a circle) of angle a :

function(A_C,r,a)=[a/2]*r^2

Area of an annular (flat ring) sector with outer radius r , innner radius s, and angle a:

function(A_A,r,s,a)=[a/2]*[r^2-s^2]

Volume of a disk-sector of radius r, angle a, and thickness T:

function(V_D,r,a,T)=[a/2]*r^2*T

Volume of the sector of a wheel of square cross-section with outer radius r , innner radius s, angle a, and thickness T :

function(V_W,r,s,a,T)=[a/2]*[r^2-s^2]*T

The edges of a rectangle with left at l, bottom at b, width w and height h.
p is the drawing parameter.

function(E,1,l,b,w,h,p)=vector(l+w*p,b,0),function(E,2,l,b,w,h,p)=vector(l+w,b+h*p,0),function(E,3,l,b,w,h,p)=vector(l+w*[1-p],b+h,0),function(E,4,l,b,w,h,p)=vector(l,b+h*[1-p],0)

Filling in the interior of the rectangle:
u is the horizontal and v is the vertical parameter.

function(I,l,b,w,h,u,v)=vector(l+w*u,b+h*v,0+0)

Plots
Dots at minimum and maximum x-values:

vector(P_(M*I*N),0,0)

vector(P_(M*A*X),0,0)

Plot the curve on the xy plane:

vector(function(P,t),function(f,function(P,t)),0)

Rotation around the x-axis by the angle a:

function(R_x,a)=matrix(3,3,1,0,0,0,cos(a),-sin(a),0,sin(a),cos(a))

Rotate the curve:

function(R_x,n*degree)*vector(function(P,t),function(f,function(P,t)),0)

The interiors of the rectangles, at start and rotated:
function(I,K*W,0,W,(K*W)^2,u,v),in(k,set(4,5,6,7,8,9,10))
function(R_x,n*degree)*function(I,K*W,0,W,(K*W)^2,u,v),in(k,set(4,5,6,7,8,9,10))

The width of the wheels:

W=0.2

The edges of the rectangles, at start:

function(E,1,k*W,0,W,function(f,k*W),t),in(k,set(-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12))

function(E,2,k*W,0,W,function(f,k*W),t),in(k,set(-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12))

function(E,3,k*W,0,W,function(f,k*W),t),in(k,set(-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12))

function(E,4,k*W,0,W,function(f,k*W),t),in(k,set(-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12))

The edges of the rectangles, rotated:

function(R_x,n*degree)*function(E,1,k*W,0,W,function(f,k*W),t),in(k,set(-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12))

function(R_x,n*degree)*function(E,2,k*W,0,W,function(f,k*W),t),in(k,set(-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12))

function(R_x,n*degree)*function(E,3,k*W,0,W,function(f,k*W),t),in(k,set(-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12))

function(R_x,n*degree)*function(E,4,k*W,0,W,function(f,k*W),t),in(k,set(-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12))

The surfaces swept out by the edges:

function(R_x,n*degree*v)*function(E,1,k*W,0,W,function(f,k*W),u),in(k,set(-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12))

function(R_x,n*degree*v)*function(E,2,k*W,0,W,function(f,k*W),u),in(k,set(-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12))

function(R_x,n*degree*v)*function(E,3,k*W,0,W,function(f,k*W),u),in(k,set(-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12))

function(R_x,n*degree*v)*function(E,4,k*W,0,W,function(f,k*W),u),in(k,set(-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12))

Graph of the formula

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