Question

In Exercises 17-22, evaluate \(\int_{S} \int f(x, y, z) d S\). $$ \begin{aligned} &f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}\\\ &S: z=\sqrt{x^{2}+y^{2}}, \quad x^{2}+y^{2} \leq 4 \end{aligned} $$

Short answer

The value of the integral is \(8\pi\).

Step-by-step solution

Change to Polar Coordinates

The equation for \(S\) suggests polar coordinates would be convenient. So, let \(x = r \cos \theta\) and \(y = r \sin \theta\). Now, the function becomes \(f(r, \theta) = r\), and the limits of the double integral become \(0 \leq r \leq 2\) and \(0 \leq \theta \leq 2\pi\).

Compute the Integral

The integral of the function over the surface now becomes \( \int_{0}^{2\pi}\int_{0}^{2} rf(r, \theta)r drd\theta\). This simplifies to \(\int_{0}^{2\pi}\int_{0}^{2} r^{3} drd\theta\).

Evaluate the Inner Integral

Evaluating the inner integral first, \(\int_{0}^{2} r^{3} dr\), we have as a result: \[\frac{1}{4}r^{4}\Big|_0^2 = 4.\]

Evaluate the Outer Integral

Now, we evaluate the outer integral, \(\int_{0}^{2\pi} 4 d\theta\), with the result: \(8\pi\).