REFER TO IMAGE # Surface Integrals and the Curl of a Vector Field## Problem Statement### Given:Let \( S \) be the part of the paraboloid \( z = 9 - x^2 - y^2 \) (with upward orientation) that lies above the plane \( z = 0 \). Let \( \mathbf{F} = \langle x, 2xz, xy \rangle \).### Tasks:(a) **Set up** an iterated integral to find \( \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \) directly.(b) **Set up** an iterated integral to find \( \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \) using an appropriate theorem.## Solution Steps### Part (a) Direct Approach1. To find the iterated integral directly, identify the surface \( S \).2. **Surface \( S \)**: - The surface is part of the paraboloid \( z = 9 - x^2 - y^2 \). - \( S \) has an upward normal.3. **Vector Field \( \mathbf{F} \)**: - \( \mathbf{F} = \langle x, 2xz, xy \rangle \).4. **Curl of \( \mathbf{F} \)**: - Evaluate \( \nabla \times \mathbf{F} \). \[ \nabla \times \mathbf{F} = \nabla \times \langle x, 2xz, xy \rangle \]5. **Surface Element \( d\mathbf{S} \)** for Paraboloid: - Calculate \( d\mathbf{S} \) for \( z = 9 - x^2 - y^2 \).6. **Integral Form**: - Set up the iterated integral involving \( (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \).### Part (b) Using an Appropriate Theorem1. **Use Stokes' Theorem**: - Stokes' Theorem relates a surface integral of the curl of a vector field to a line integral around the boundary curve of the surface. \[ \iint_S (\

Name: REFER TO IMAGE # Surface Integrals and the Curl of a Vector Field## Pr
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Description: REFER TO IMAGE # Surface Integrals and the Curl of a Vector Field## Problem Statement### Given:Let \( S \) be the part of the paraboloid \( z = 9 - x^2 - y^2 \) (with upward orientation) that lies above the plane \( z = 0 \). Let \( \mathbf{F} = \lan