Evaluate the Surface Integral S of the Paraboloid y = x² + z² that Lies Inside the Cylinder x² + z² = 1

To evaluate the surface integral of the given paraboloid that lies within the specified cylindrical region, we first need to parameterize the surface and set up the integral appropriately.

The surface is defined by the equation of a paraboloid: y = x² + z². We also have a constraint given by the cylinder: x² + z² = 1.

We can use polar coordinates to parameterize the surface within the constraints of the cylinder:

• Let x = r cos(θ)
• Let z = r sin(θ)
• Where 0 ≤ r ≤ 1 and 0 ≤ θ < 2π

This parameterization gives us:

• y = r²

The surface can thus be expressed in the form:

• S(r, θ) = (r cos(θ), r², r sin(θ))

Next, we need to calculate the surface area element (dS) for our parameterization. To do this, we can compute the tangent vectors:

• S_r = (cos(θ), 2r, sin(θ))
• S_θ = (-r sin(θ), 0, r cos(θ))

The cross product of these tangent vectors will give us the normal vector to the surface:

N = S_r × S_θ

Calculating the cross product:

• N = (2r^2 cos(θ), -r, 2r^2 sin(θ))

The magnitude of this normal vector is:

• |N| = √((2r² cos(θ))² + (-r)² + (2r² sin(θ))²) = √(4r^4 + r^2) = r√(4r² + 1)

Now we can express the surface integral of a function f(x, y, z) over the surface S:

∬_S f(x, y, z) dS = ∬_D f(r cos(θ), r², r sin(θ)) |N| dr dθ

Substituting the variables defined for r and θ, we can set our limits of integration:

• 0 ≤ r ≤ 1
• 0 ≤ θ < 2π

Putting everything together, we evaluate the integral by replacing f(x, y, z) as needed, and integrating over the defined region. This will yield the desired surface integral value.