7-32. Consider a circle of radius ρ < a centered at x=a > 0 in the xz plane. The
parametric equations of this circle are
x−a=ρcos θ, z =ρsin θ, 0 ≤θ≤ 2 π, a > ρ.If the circle is rotated about the z-axis, a torus results.
(a) Show that the parametric equations of the torus are
x= (a+ρcos θ) cos φ, y = (a+ρcos θ) sin φ, z =ρsin θ, 0 ≤φ≤ 2 π(b) Find the surface area of the torus.
(c) Find the volume of the torus.
7-33. Calculate the arc length along the given curve between the points specified.
(a) y=x, p 1 (0 ,0), p 2 (3,3)
(b) x= cos t, y = sin t, 0 ≤t≤ 2 π
(c) x=t, y = 2t, z = 2t, p 1 (0 , 0 ,0), p 2 (2 , 4 ,4)
(d) y=x^2 , p 1 (0 ,0), p 2 (2 ,4)7-34.
(a) Describe the surface r =uˆe 1 +vˆe 2 and sketch some coordinate curves on the
surface.
(b) Describe the surface r =vcos uˆe 1 +vsin uˆe 2 and sketch some coordinate curves on
the surface.
(c) Describe the surface r = sin ucos vˆe 1 + sin usin vˆe 2 + cos uˆe 3 and sketch some coor-
dinate curves on the surface.
(d) Construct a unit normal vector to each of the above surfaces.
7-35. Evaluate the surface integral I=
∫∫SF·dS , where F= 4yˆe 1 + 4(x+z)ˆe 2 and
Sis the surface of the plane x+y+z= 1 lying in the first octant.
7-36. Evaluate the surface integral I=
∫∫SF·dS, where F =x^2 ˆe 1 +y^2 ˆe 2 +z^2 ˆe 3
and S is the surface of the unit cube bounded by the planes x= 0, y = 0, z = 0 and
x= 1, y = 1, z = 1.