Question5. Find $\iint_{S}(x+z) \mathrm{dS}, \mathrm{S}$ : the first octant portion of the cylinder $y^{2}+z^{2}=9$ between $x=0$ and $x=4$ i) Sketch S ii) Use the parametric representation method to evaluate this.

Studdy Solution
Evaluate the surface integral.
- Evaluate the inner integral with respect to $\theta$: \int_{0}^{\frac{\pi}{2}} (x + 3\sin\theta) \cdot 3 \, \mathrm{d}\theta = \int_{0}^{\frac{\pi}{2}} (3x + 9\sin\theta) \, \mathrm{d}\theta \] = 3x \left[ \theta \right]_{0}^{\frac{\pi}{2}} + 9 \left[ -\cos\theta \right]_{0}^{\frac{\pi}{2}} \] = 3x \cdot \frac{\pi}{2} + 9(0 + 1) \] = \frac{3\pi}{2}x + 9 \]
- Evaluate the outer integral with respect to $x$: \int_{0}^{4} \left( \frac{3\pi}{2}x + 9 \right) \, \mathrm{d}x \] = \left[ \frac{3\pi}{4}x^2 + 9x \right]_{0}^{4} \] = \left( \frac{3\pi}{4} \cdot 16 + 36 \right) - \left( 0 + 0 \right) \] = 12\pi + 36 \]
The value of the surface integral is:
$$\boxed{12\pi + 36}$$