Evaluate the surface integral. ∬S y^2 d S S is the part of the sphere x^2+y^2+z^2=4

Evaluate the surface integral. ∬S y^2 d S S is the...

Key Concepts

Surface Integral

A surface integral extends the concept of an integral to functions defined on a surface. It is used to sum a function over a curved, two-dimensional manifold, and often involves integrating a scalar function multiplied by the differential area element over that surface.

Parameterization of Surfaces

Parameterization is a method to describe a surface using two parameters. For surfaces like spheres, choosing appropriate parameters simplifies the evaluation of the integral, as it allows the surface element to be expressed in terms of these parameters, making the integration process more manageable.

Differential Surface Element

The differential surface element, often denoted as dS, represents an infinitesimally small piece of area on the surface. In many cases, dS is expressed in terms of the parameters used to describe the surface, capturing how the area stretches or contracts due to the parametrization.

Spherical Coordinates

Spherical coordinates are a coordinate system particularly well-suited for dealing with problems involving spheres. They simplify the representation and integration over spherical surfaces by naturally aligning with the symmetry of the sphere and providing clear expressions for the points and area elements on the sphere.

Domain Restriction via Intersection

When a surface is limited by another geometric object, such as a cylinder intersecting a sphere, the domain of integration is restricted accordingly. This requires determining the appropriate limits of the parameters to accurately represent the portion of the surface over which the integration is performed.

Transcript

00:01 So we are given that x squared plus y squared plus z squared is equal to four.

00:09 So one way to do this problem is we can solve for z.

00:14 In this case, i will give that z is equal to four minus x squared minus y squared, the square root of those three numbers.

00:25 It's not plus or minus in this case because the question specifically asks that it is above the xy plane.

00:36 And since now that we have z in terms of x and y, we can use the formula that the surface integral of f of x, y, g of x, y, we can use that this multiplied by the square roots of the z partial with respect to x squared plus the z partial with respect to y squared plus 1, da, we can use this formula to evaluate the integral.

01:09 So going farther with this, the x and y partial are not going to be extremely pretty, as z is equal to a square root of both of them.

01:22 But when worked out, this will give that the x partial of d z is equal to 1 over 2 square root 4 minus x squared minus y squared, multiplied by 2x, negative 2x.

01:39 And the y partial will be the same but with a negative 2y that is that that is equal to 1 over 2 square 2x 4 minus x squared minus y squared multiply by negative 2y and both of these then will become equal to negative x over square root 4 minus x squared minus y squared and negative y over square root 4 minus x squared minus y squared.

02:23 Now that we have that, we can set up our equation as this will be equal to the integral of y multiplied by 2 over square root 4 minus x squared minus y squared d a...

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