CALCULUS INTEGRALS
  1. CALCULUS ELLIPSE CONE:
      height = h with base (x/a)2 + (y/b)2 = 1  
    cone1.gif
    From similar triangles:
    x/a = (h - z)/h y/b = (h - z)/h
    x = a(h - z)/h y = b(h - z)/h
    Area = xy = ab(h - z)2/h2
    Volume = ab/h2(h - z)2dz
    = ab/h2(h2 - 2hz + z2)dz
    = ab/h2(h2z - hz2 + z3/3) = abh/3
    with a = b = r Volume = r2h/3
     
      ANOTHER METHOD FOR A REGULAR CONE: using dz 
    cone1h.gif
    r/R = (h - z)/h or r = R(h - z)/h
    dr/dz = -R/h or dr= -(R/h)dz
    Area = (r*dz - dr*dz/2) = R/h((h - z)*dz + dz2/2)
    Volume = Area*ds = Area*r*d
    Volume = Area*ds = R/h[(h - z)*dz]r*d
    Volume = (R/h)2[(h - z)2*dz]d
    Volume = (R/h)2(h - z)2*dz
    Volume = (R/h)2[(h - z)3/3]
    Volume = R2h/3
     
      ANOTHER METHOD FOR A REGULAR CONE: using dr 
    cone1hr.gif
    r/R = (h - z)/h or r = R(h - z)/h
    dr/dz = -R/h or dr= -(R/h)dz or dz= -(h/R)dr
    Area = (r*dz - dr*dz/2) = (r - dr/2)*dz = -(h/R)(r - dr/2)*dr
    Volume = Area*ds = (r*dz - dr*dz/2)*r*d
    Volume = [(r2*dr - r*dr*dz/2)]d
    Volume = 2(R/h)[(R/h)(h - z)2 - (h - z)*dr/2]*dz
    Volume = -(R/h)[(R/h)(h - z)3/(-3) - (h*z - z2)*dr/2]
    Volume = -(R/h)2*h3/(-3)
    Volume = R2h/3

  2. PARABOLOID:
    f(p,0,z) = p2 = 9 - z
    cone2.gif
    Triple integral for entire surface:
    = p2dV = p2(dz dp p*d)
    = p3(9 - p2) dp d
    = [ (9/4)p4 - (1/6)p6] d
    = (1/12)[ (27)p4 - (2)p6] d
    = (729/12)d = 243/2

  3. PARABOLOID:
    x2 + 4y2 = z
    cylinders y2 = x & x2 = y
    cone3.gif
    Comon volume of 3 cylinders above x0y plane:
    Volume = z*dx*dy = (x2 + 4y2)dx*dy
    Volume = (x2y + 4y3/3)*dx
    Volume = [(x2.5 + 4x1.5/3) - (x4 + 4x6/3)]dx
    Volume = [x3.5/3.5 + 4x2.5/7.5 - x5/5 - 4x7/21]
    Volume = [2/7 + 8/15 - 3/15 - 4/21] = 2/21 + 5/15
    Volume = 135/315 = 27/63 = 3/7 cubic units

  4. CALCULUS SPHERE:
    4/3R3
    cone4.gif
    Volume of a sphere:
    Volume = dV = ds*ds1*ds2
    Volume = (p*d)(psin*d)(dp)
    Volume = p2sin*dp*d*d
    Volume = (1/3)R3sin*d*d
    Volume = (1/3)R3sin*d
    Volume = (4/3)R3(-cos)sinb0290.gif
    Volume = 4R3/3


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