Find the area of the surface.
The part of the paraboloid
z = x2 + y2
that lies inside the cylinder
x2 + y2 = 4.
Its wrong Steve
Odd. It agrees with the formula here
http://www.had2know.com/academics/paraboloid-surface-area-volume-calculator.html
Better double-check my numbers.
To find the area of the surface that lies inside the cylinder, we can use a method called surface integral.
First, let's parameterize the surface of the paraboloid. We can use cylindrical coordinates to do this. The conversion from Cartesian coordinates (x, y, z) to cylindrical coordinates (ρ, θ, z) is given by:
x = ρcos(θ)
y = ρsin(θ)
z = z
In this case, since the paraboloid is symmetric with respect to the z-axis, we can ignore the angle θ and only consider the radial distance ρ and the height z.
Next, we need to determine the limits of integration. Since the paraboloid lies inside the cylinder x^2 + y^2 = 4, we know that ρ can range from 0 to 2 (the square root of 4). The height z can range from the bottom of the paraboloid to the top.
The bottom of the paraboloid is given by z = x^2 + y^2. Substituting the expressions for x and y in terms of ρ, we get z = ρ^2. The top of the paraboloid is given by z = 4, since the cylinder intersects the paraboloid at z = 4 for all values of ρ.
Now, we can set up the surface integral to calculate the area. The area is given by:
A = ∫∫√((∂z/∂ρ)^2 + (∂z/∂z)^2 + 1) dA
where dA is the element of surface area in cylindrical coordinates, given by dA = ρ dρ dz.
Substituting the expressions for z in terms of ρ, the integrand simplifies to:
√((2ρ)^2 + 1)
Evaluating the double integral using the limits mentioned above will give us the area of the surface that lies inside the cylinder.
Note: The calculations for the surface integral can be quite involved. It might be easier to use computer software or numerical methods to evaluate the integral.
In cylindrical coordinates, you have
z = r^2
r = 2
symmetry allows us to say
v = 4∫[0,2]∫[0,π/2] r^2 * r dr dθ
= 4∫[0,2]∫[0,π/2] r^3 dr dθ
= 2π ∫[0,2] r^3 dr
= 2π * 16/4
= 8π