(a) Find an equation of the tangent plane at the point ( 4 , − 2 , 1 ) to the parametric surface S given by r ( u , v ) = v 2 i − u v j + u 2 k 0 ⩽ u ⩽ 3 , − 3 ⩽ v ⩽ 3 (b) Graph the surface S and the tangent plane found in part (a). (c) Set up, but do not evaluate, an integral for the surface area of S . (d) If F ( x , y , z ) = z 2 1 + x 2 i + x 2 1 + y 2 j + y 2 1 + z 2 k use a computer algebra system to find ∬ S F ⋅ d S correct to four decimal places.
(a) Find an equation of the tangent plane at the point ( 4 , − 2 , 1 ) to the parametric surface S given by r ( u , v ) = v 2 i − u v j + u 2 k 0 ⩽ u ⩽ 3 , − 3 ⩽ v ⩽ 3 (b) Graph the surface S and the tangent plane found in part (a). (c) Set up, but do not evaluate, an integral for the surface area of S . (d) If F ( x , y , z ) = z 2 1 + x 2 i + x 2 1 + y 2 j + y 2 1 + z 2 k use a computer algebra system to find ∬ S F ⋅ d S correct to four decimal places.
Solution Summary: The author explains how to calculate the tangent vectors by differentiating the parametric surface equation.
(a) Find an equation of the tangent plane at the point
(
4
,
−
2
,
1
)
to the parametric surface
S
given by
r
(
u
,
v
)
=
v
2
i
−
u
v
j
+
u
2
k
0
⩽
u
⩽
3
,
−
3
⩽
v
⩽
3
(b) Graph the surface
S
and the tangent plane found in part (a).
(c) Set up, but do not evaluate, an integral for the surface area of
S
.
(d) If
F
(
x
,
y
,
z
)
=
z
2
1
+
x
2
i
+
x
2
1
+
y
2
j
+
y
2
1
+
z
2
k
use a computer algebra system to find
∬
S
F
⋅
d
S
correct to four decimal places.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
How do you calculate the area of a parametrized surface in space? Of an implicitly defined surface F(x, y, z) = 0? Of the surface which is the graph of z = ƒ(x, y)? Give examples.
1. Find the centroid of the region bounded by
y = x^3 , y = −x + 2, y = 0.
2. Find an equation of the tangent line to the curve whose parametric equation is
x = tan θ, y = sec θ
Consider the surface x2 + y2 − 2xy − x + 4y − z2 + 4 = 0 at the point Po (2, −2, 5).
Determine the following:(a) the general equation of the tangent plane to the surface at Po(b) the parametric equations of the normal line to the surface at Po
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.
Area Between The Curve Problem No 1 - Applications Of Definite Integration - Diploma Maths II; Author: Ekeeda;https://www.youtube.com/watch?v=q3ZU0GnGaxA;License: Standard YouTube License, CC-BY