Find the following average values. The average z -coordinate...

Key Concepts

Volume Integration in Multivariable Calculus

Volume integration is the process of integrating a multivariable function over a three-dimensional region. This technique requires setting up an appropriate triple integral, often using coordinate transformations, to account for the limits and geometry of the region involved.

Spherical Coordinates

Spherical coordinates are a system of curvilinear coordinates that are particularly useful for regions exhibiting spherical symmetry. In spherical coordinates, points in space are described by a radius, a polar angle, and an azimuthal angle, which simplifies the integration over spherical or hemispherical regions due to the alignment with the geometry.

Average Value of a Function Over a Region

This concept involves computing the average value of a function defined over a spatial region by integrating the function over the region and dividing by the volume (or area, in two dimensions) of the region. It is a fundamental idea in calculus and physics where notions of mean or expectation over continuous domains are required.

Use of the Jacobian in Coordinate Transformations

When changing from Cartesian to spherical coordinates, the Jacobian determinant accounts for the distortion in volume element due to the transformation. This is essential in ensuring that the integral in the new coordinates accurately represents the true volume of the region.

Transcript

00:01 So we are given this hemisphere with a radius of 4, it's based in the xy plane, and its center at the origin, and i'll define the average value of the z coordinates within this hemisphere.

00:12 So we know that the average value formula is equal to fbar, equal 1 over the volume of the solid, times the triple integral over d of the function f of x, y, and z, d.

00:33 So let's start by finding d, which we'll use to evaluate this integral.

00:38 So we know that the equation of a full sphere with radius 4 would be x squared plus y squared plus z squared equals 16, because the center of the sphere is at the origin.

00:54 So for a hemisphere, we will take z only in this positive half.

01:02 So we'll go from 0 to root 16 minus x squared minus y squared.

01:09 Let's write that down.

01:17 So we'll take 0 to the positive square root of 4 squared which is 16 minus x squared minus y squared.

01:31 And then if we look at the projection of this hemisphere into the xy plane, it is a circle centered at the origin with radius 4.

01:39 So we can write that for y.

01:43 We'll take values from the negative root 4 squared minus x squared to positive root 4 squared minus x squared.

01:54 So we'll have negative root 16 minus x squared, less than or equal to y, less than or equal to positive root 16 minus x squared.

02:03 And we will take x from negative 4 to positive 4 because 4 is the radius of the circle.

02:16 All right.

02:18 Now let's set up this integral and then after that we'll find the volume and combine them to find the average value.

02:26 So the integral over d of f of x of y of z d is equal to.

02:35 We'll put x on the very outside, y as the middle integral, and then z on the very inside.

02:43 So the integral from negative 4 to positive 4, the integral from negative root 16 minus x squared to positive root 16 minus x squared of the integral from 0 to root 16 minus x squared minus y squared of f of x of y of z so because we're asked to find the average value of the z coordinate this function is just going to be z d z d y d x d x d x all right now we can evaluate this integral of z with respect to z so we can copy over the first two integration symbols again the integral of z will be one half z squared and we'll evaluate that from z equals 0 to z equals root 16 minus x squared minus y squared do y d x so this is equal to the integral from negative 4 to 4 of the integral from negative root 16 minus x squared to positive root 16 minus x squared of one half times the square root of 16 minus x squared minus y squared quantity squared which is 16 minus x squared minus y squared minus 1 half times zero squared which is just zero d.

04:38 Y dx...

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