Section 5.2 Examples
Calculus I
Andy Long

#48, p. 361

Clear[r,R,a,x,y,z]

V=Integrate[Pi(R+z((r-R)/h))^2,{z,0,h}]

#52, p. 361

V=Integrate[(a-3Sqrt[3]/2z)^2*Sqrt[3]/4,{z,0,2*a/(3Sqrt[3])}]
(* if it's not a regular tetrahedron, then *)
V=Integrate[Sqrt[3]/4*a^2(1-z/h)^2,{z,0,h}]

#4, p. 360

Plot[{Sqrt[25-x^2]},{x,0,5}]
Plot[{Sqrt[25-x^2],-Sqrt[25-x^2]},{x,0,5}]
V=Integrate[Pi(Sqrt[25-x^2])^2,{x,2,4}]

#5, p. 360

Plot[{(x/2)^2,9},{x,0,6}]
Plot[{(x/2)^2,9},{x,-6,6}]
V=Integrate[Pi(2*Sqrt[y])^2,{y,0,9}]
(* what if we rotated about the x-axis instead? *)

Plot[{(x/2)^2,9,-(x/2)^2,-9},{x,0,6}]
(* what if we rotated about the x-axis instead? *)
V=Integrate[Pi(9^2-((x/2)^2)^2),{x,0,6}]

Cube:

Clear[r,R,a,x,y,z,s,h]

V=Integrate[s^2,{x,0,s}]

Cylinder:

V=Integrate[ Pi r^2,{x,0,h}]

Cone:

V=Integrate[Pi (r(1-x/h))^2,{x,0,h}]

Pyramid:

V=Integrate[(s(1-x/h))^2,{x,0,h}]

Sphere:

V=2 Integrate[Pi(Sqrt[r^2-x^2])^2,{x,0,r}]