Chapter 1. calc_tutorial_16_4_019

1.1 Problem Statement

{3,5,7,9}
pow($a,2)
{2,3,4}
2*$a

Calculate for the given surface and function.

x2 + y2 = $b, 0 ≤ z ≤ $c; f(x, y, z) = e−z

1.2 Step 1

Question Sequence

Question 1.1

Recall how to find surface integrals with a parametrized surface. Let G(u, v) be a parametrization of a surface S with parameter domain D. Assume that G is continuously differentiable, one-to-one, and regular (except possibly at the boundary of D). Then we can calculate the integral of a function over this surface.

Choose the correct integration for below.

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

In order to use this equation, we must parametrize the surface described by x2 + y2 = $b, 0 ≤ z ≤ $c.

The given surface represents a portion of a cylinder, so it will be easiest to integrate if we parametrize with 5EwKpZPI+Ks+8IhwDXS6p3pcxi0QKDe/pNP6WQ== coordinates.

In the coordinates (θ, z), we have the following parametrization of the surface.

Φ(θ, z) = (nc1ItEz0kR4=cos θ, nc1ItEz0kR4=sin θ, z)

The bounds on z and θ are 0 ≤ z ≤ SFgqQUkJGdg= and 0 ≤ θ ≤ XvVM00l89Is=π.

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

Compute Tθ and Tz, the tangent vectors to the surface Φ(θ, z) = ($a cos θ, $a sin θ, z).

Tθ = (∂ / ∂θ)($a cos θ, $a sin θ, z)

= <IHLqJALES1Y=sin θ, nc1ItEz0kR4=cos θ, 0>

Tz = (∂ / ∂z)($a cos θ, $a sin θ, z)

= <0, 0, 0VV1JcqyBrI=>

Use these vectors to find n(θ, z), the normal vector to the surface. Let the table below represent the matrix A.

i j k
-$a sin θ $a cos θ 0
0 0 1
Table : Matrix A

n(θ, z) = Tθ × Tz

= det(A)

= <nc1ItEz0kR4=cos θ, nc1ItEz0kR4=sin θ, 0>

Correct.
Incorrect.

1.4 Step 3

Question Sequence

Question 1.3

Compute the length of the normal vector, ||n(θ, z)||.

||n(θ, z)|| = ||<$a cos θ, $a sin θ, 0>||

= √(($a cos θ)2 + (nc1ItEz0kR4=sin θ)2)

= nc1ItEz0kR4=

Correct.
Incorrect.

1.5 Step 4

Question Sequence

Question 1.4

Compute f(Φ(θ, z)) given f(x, y, z) = e−z and Φ(θ, z) = ($a cos θ, $a sin θ, z).

f(Φ(θ, z)) =

QmIOronH8211WXLtolxCAE94DMAGO4BrTbEuiUPsNjhaNrADWLfZCNOGSe4K4+PTdEYOkEvKSaGgP/9S9h6gU5ahFbNTGYfIYUrOr2qjf73h3f8ofWidnYETeGAapT49Wb8+lidX0jczcx5XGtxK6USg1Hdy3qDIae79Qw==
Correct.
Incorrect.

1.6 Step 5

Question Sequence

Question 1.5

Calculate for the surface described by x2 + y2 = $b, 0 ≤ z ≤ $c, and the function f(x, y, z) = e−z.

= U2GIbglD1oM=π(1 - e-$c)

Correct.
Incorrect.