33. a. Parametrization of an ellipsoid The parametrization x = a cos 0, y = b sin 0, 0 < 0 < 27 gives the ellipse (x²/a²) + (y²/b²) = 1. Using the angles 0 and ø in spheri- cal coordinates, show that r(0, 4) = (a cos 0 sin ø)i + (b sin 0 sin ø)j + (c cos 4)k is a parametrization of the ellipsoid (x²/a²) + (y²/b²) + (z²/c²) = 1. b. Write an integral for the surface area of the ellipsoid, but do not evaluate the integral.

44. The parametric equation of the rectangular hyperbola #(y + 1) = c² is A. 1 = ct, y B. I = 1+ ct, C. z = ct, y = - I = D. I = ct, y =

Parametrize the intersection of the surfaces y² - z² = x - 6, y² + z² = 25 using trigonometric functions. (Express numbers in exact form. Use symbolic notation and fractions where needed. Give the parametrization of the y variable in the form a cos (t).) x(t) = y(t) = z(t) =

(a) Find three parametrizations that give the same graph as the given equation. (b) Find three parametrizations that give only a portion of the graph of the given equation.y = x2