(a) Show that the parametric equations x = a sin u cos v , y = b sin u sin v , z = c cos u , 0 ≤ u ≤ π , 0 ≤ v ≤ 2 π , represent an ellipsoid. (b) Use the parametric equations in part (a) to graph the ellipsoid for the case a = 1 , b = 2 , c = 3 (c) Set up, but do not evaluate, a double integral for the surface area of the ellipsoid in part (b).
(a) Show that the parametric equations
x
=
a
sin
u
cos
v
,
y
=
b
sin
u
sin
v
,
z
=
c
cos
u
,
0
≤
u
≤
π
,
0
≤
v
≤
2
π
, represent an ellipsoid.
(b) Use the parametric equations in part (a) to graph the ellipsoid for the case
a
=
1
,
b
=
2
,
c
=
3
(c) Set up, but do not evaluate, a double integral for the surface area of the ellipsoid in part (b).
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Show that the parametric equations x =a sin u cos v, y =b sin u sin v z=c cos u 0<u <pi 0 v 2pi represent an ellipsoid.
7. a. Find the parametric equations for the surface generated byrevolving the curve y = sin x about the x-axis.
b. Using the parametric equations from part a. set up but do NOTevaluate an integral that will give the surface area of that portion ofthe surface for which 0 ≤ x ≤ π.
c. . Find the equation of the tangent plane to the parametric surfacein part a. at the point (x, y, z) = (pi/6, 1/2, 0)
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.