Practice problem taken from Denny Gulick, Robert Ellis - Calculus With Concepts in Calculus. Sixth-Cengage Learning, chapter 15.6. Requesting help with number 12.
Please show and explain ALL steps; I'm trying to practice for an exam and need help understanding this. Do not use Stokes Theorem, or Divergence Theorem, as that's not covered until the following 2 chapters.
Transcribed Image Text:In Exercises 10-12 use (4) or (6) to evaluate ſſ F · n dS. 10. F(x, y, z) = yi − xj + 8k; Σ is the part of the paraboloid z=9-x²² above the xy plane; n is directed upward. 11. F(x, y, z) = xi − j + 2x²k; Σ is the part of the paraboloid z = x² + 12² above the region in the xy plane bounded by the parabolas x = 1 − y² and x = y² – 1; n is directed downward. - 12. F(x, y, z) = xi + yj + zk; Σ is the cube with vertices (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), and (1, 1, 1); n is directed outward from the cube.

Question

Practice problem taken from Denny Gulick, Robert Ellis - Calculus With Concepts in Calculus. Sixth-Cengage Learning, chapter 15.6. Requesting help with number 12.
Please show and explain ALL steps; I'm trying to practice for an exam and need help understanding this. Do not use Stokes Theorem, or Divergence Theorem, as that's not covered until the following 2 chapters.

In Exercises 10-12 use (4) or (6) to evaluate ſſ F · n dS.
10. F(x, y, z) = yi − xj + 8k; Σ is the part of the paraboloid
z=9-x²² above the xy plane; n is directed upward.
11. F(x, y, z) = xi − j + 2x²k; Σ is the part of the paraboloid
z = x² + 12² above the region in the xy plane bounded by
the parabolas x = 1 − y² and x = y² – 1; n is directed
downward.
-
12. F(x, y, z) = xi + yj + zk; Σ is the cube with vertices (0, 0, 0),
(0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), and
(1, 1, 1); n is directed outward from the cube.

Accepted Answer

Step 1:1. Parametrize each face of the cube:We will parametrize each face of the cube and determine…