Use spherical coordinates. Evaluate ∭E y^2 d V, where E is the solid hemisphere x^2+y^2+z^2 ⩽9,

Use spherical coordinates. Evaluate ∭E y^2 d V, where E is...

Key Concepts

Conversion of the Integrand

When changing from Cartesian to spherical coordinates, each occurrence of a Cartesian variable must be replaced with its spherical equivalent. For instance, the integrand y² transforms to (? sin(?) sin(?))². This conversion is critical to correctly reformulate the integral in the new coordinate system before carrying out the integration.

Limits of Integration for Hemispherical Regions

Defining the limits of integration is crucial when the region of interest is a hemisphere. For a sphere of radius R, ? ranges from 0 to R and ? typically ranges from 0 to ?. However, additional constraints such as y ? 0 require careful adjustment of the limits on ? (or sometimes ?) so that only the appropriate half of the sphere is integrated over. Correctly setting these limits ensures the integration covers the desired region.

Spherical Coordinate System

Spherical coordinates are a three-dimensional coordinate system where points are represented by a distance from the origin (?), a polar angle (?) measured from the positive z-axis, and an azimuthal angle (?) measured in the xy-plane. This coordinate system is particularly useful for regions with spherical symmetry or boundaries defined by spheres, making integration over spherical regions more straightforward.

Jacobian in Spherical Coordinates

When converting a triple integral from Cartesian to spherical coordinates, the volume element changes to include the Jacobian determinant, which in spherical coordinates is ?² sin(?). This factor accounts for the distortion in volume when mapping from Cartesian to spherical coordinates and is essential to correctly evaluate the integral.

Transcript

00:03 In this video, we are going to evaluate an integral using spherical coordinates.

00:10 We're integrating over a specified hemispheres, and we have just a specified region.

00:17 So, why don't we first try and convert everything all the way over to our spherical coordinates? now, we have to remember our spherical conversions.

00:28 We remember that x equals row times the sign of 5.

00:36 Times the cosine of theta.

00:40 We also know that y equals row times the sign of phi times the sign of theta.

00:48 We know that z equals row times the cosine of phi.

00:56 Furthermore, we know that x squared plus y squared plus z squared equals row squared.

01:11 And we know that our differential volume is going to be equal to row squared times the sign of phi times d row, d theta, d -fi.

01:23 Okay.

01:25 Now, with these conversions in mind, let's start by examining our region.

01:31 I can rewrite this region as being row squared is less than or equal to nine, which means that row has to be less than equal to three itself.

01:44 So we're going to be looking at, so we're actually going to be integrating over a sphere with a radius of three.

01:54 Now, the second step is what i'm going to do is i'm going to project.

01:58 This region onto the xy plane.

02:05 So our region is looking something like this because we're given that our y value has to be strictly positive...

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