11.6: Surface Integrals

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May 4, 2024
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- 11.5: Green's Theorem
- 11.7: Stokes' Theorem

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Find a parametric description for the surface $3x - 2y + z = 2$ .
Find a parametric description for the surface $z = x^2 + y^2$ .
Find a parametric description for the surface $z^2 = x^2 + y^2$ for $2 \leq z \leq 8$ .
Find a parametric description for the surface $x^2 + y^2 = 9$ .
Find a parametric description for the surface $x^2 + y^2 + z^2 = 4$ .
Find a parametric description for the portion of the sphere $x^2 + y^2 + z^2 = 4$ below the plane $z = 1$ .
Let $\sigma$ be the portion of the sphere $x^2 + y^2 + z^2 = 4$ with $z \geq 0$ . Evaluate $\displaystyle \iint_\sigma z\ dS$ .
Let $\sigma$ be the portion of the cone $z = \sqrt{x^2 + y^2}$ between $z = 1$ and $z = 3$ . Evaluate $\displaystyle \iint_\sigma z^2\ dS$ .
Let $\sigma$ be the portion of the plane $x + y + z = 1$ in the first octant. Evaluate $\displaystyle \iint_\sigma xy\ dS$ .
Evaluate $\displaystyle \iint_\sigma (2, -3, 4) \cdot \mathbf{N}\ dS$ , where $\sigma$ is the paraboloid $z = x^2 + y^2$ with $-1 \leq x \leq 1$ , $-1 \leq y \leq 1$ , oriented up.
Evaluate $\displaystyle \iint_\sigma (x, y, -2) \cdot d\mathbf{S}$ , where $\sigma$ is given by $z = 1 - x^2 - y^2$ , where $x^2 + y^2 \leq 1$ , oriented up.
Evaluate $\displaystyle \iint_\sigma (xy, yz, zx) \cdot d\mathbf{S}$ , where $\sigma$ is the paraboloid $z = x + y^2 + 2$ , where $0 \leq x \leq 1$ , $x \leq y \leq 1$ , oriented up.

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