Question

Evaluate the surface integral. ??s (x2 y2 z2 ds s is the part of the cylinder x2 y2 = 4 that lies between the planes z = 0 and z = 2, together with its top and bottom disks.

Answer 1

To evaluate the surface integral of the given cylinder, you can divide it into three parts: the curved surface, and the two circular top and bottom disks. First, calculate the surface integral for the curved surface by parameterizing it and converting it to a double integral. Then, calculate the surface integrals for the top and bottom disks using the respective equations and parameterizations.

Step-by-step explanation:

To evaluate the surface integral of the given cylinder, we can divide it into three parts: the curved surface, and the two circular top and bottom disks. Let's first calculate the surface integral for the curved surface.

Using the equation of the cylinder, we have x^2 + y^2 = 4. We can parameterize it as x = 2cos(t), y = 2sin(t) and z = z, where 0 <= t <= 2π and 0 <= z <= 2.

The surface integral of the curved surface is given by ∬ds = ∬√(1+f'x^2+f'y^2) dA, where f(x,y,z) = x^2 + y^2 = 4 and dA is the infinitesimal area element on the xy-plane.

We can evaluate the surface integral by converting it to a double integral in terms of t and z and calculating the integral.

Next, we calculate the surface integral for the top disk. The equation of the top disk is z = 2 and its radius is 2. We can parameterize it as x = rcos(t), y = rsin(t) and z = 2, where 0 <= t <= 2π and 0 <= r <= 2.

Similarly, we calculate the surface integral for the bottom disk, which has the equation of z = 0 and the same radius as the top disk. We can use the same parameterization as the top disk.

Answer 2

The surface
`S` can be parameterized by


`\mathbf r(\theta,z)=(2\cos\theta,2\sin\theta,z)`

where
`0\le\theta\le2\pi` and
`0\le z\le2`. Then the surface integral can be computed with


`\displaystyle\iint_S(x^2+y^2+z^2)\,\mathrm dS`

`\displaystyle=\int_(\theta=0)^(\theta=2\pi)\int_(z=0)^(z=2)(4\cos^2\theta+4\sin^2\theta+z^2)\|\mathbf r_\theta*\mathbf r_z\|\,\mathrm dz\,\mathrm d\theta=\frac{128\pi}3`