calc_tutorial_16_4_019

Question 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.

A.

B.

C.

In order to use this equation, we must parametrize the surface described by x² + y² = 9, 0 ≤ z ≤ 2.

The given surface represents a portion of a cylinder, so it will be easiest to integrate if we parametrize with --------cylindricalspherical coordinates.

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

Φ(θ, z) = (cos θ, sin θ, z)

The bounds on z and θ are 0 ≤ z ≤ and 0 ≤ θ ≤ π.

Question 2

Compute T_θ and T_z, the tangent vectors to the surface Φ(θ, z) = (3 cos θ, 3 sin θ, z).

T_θ = (∂ / ∂θ)(3 cos θ, 3 sin θ, z)

= <sin θ, cos θ, 0>

T_z = (∂ / ∂z)(3 cos θ, 3 sin θ, z)

= <0, 0, >

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

n(θ, z) = T_θ × T_z

= det(A)

= <cos θ, sin θ, 0>

Question 3

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

||n(θ, z)|| = ||<3 cos θ, 3 sin θ, 0>||

= √((3 cos θ)² + (sin θ)²)

=

Question 4

Compute f(Φ(θ, z)) given f(x, y, z) = e^−z and Φ(θ, z) = (3 cos θ, 3 sin θ, z).

f(Φ(θ, z)) =

A.

e−z cos θ

B.

3 cos θ

C.

e−z

D.

3 sin θ

Question 5

Calculate for the surface described by x² + y² = 9, 0 ≤ z ≤ 2, and the function f(x, y, z) = e^−z.

= π(1 - e^-2)