In the following exercises, the function f and region E are...

In the following exercises, the function $f$ and region $E$ are given. a. Express the region $E$ and the function $f$ in cylindrical coordinates. b. Convert the integral $\iiint_{B} f(x, y, z) d V \quad$ into cylindrical coordinates and evaluate it. $f(x, y, z)=x^{2}+y^{2}$ $E=\left\{(x, y, z) | 0 \leq x^{2}+y^{2} \leq 4, y \geq 0,0 \leq z \leq 3-x\right\}$

In the following exercises, the function $f$ and region $E$ are given.
a. Express the region $E$ and the function $f$ in cylindrical coordinates.
b. Convert the integral $\iiint_{B} f(x, y, z) d V \quad$ into cylindrical coordinates and evaluate it.
$f(x, y, z)=x^{2}+y^{2}$
$E=\left\{(x, y, z) | 0 \leq x^{2}+y^{2} \leq 4, y \geq 0,0 \leq z \leq 3-x\right\}$

Key Concepts

Triple Integration Setup

Setting up a triple integral in cylindrical coordinates involves expressing the integrand, region, and differential volume element in terms of r, ?, and z. This often simplifies the integration if the function and the region have circular symmetry. The integration limits are determined from the converted inequalities, and the integration is performed step-by-step over r, ?, and z.

Jacobian Determinant

In changing variables from Cartesian to cylindrical coordinates, it is necessary to include the Jacobian determinant, which in this case is r. This factor accounts for the change in the volume element such that the differential volume dV in Cartesian coordinates becomes r dr d? dz in cylindrical coordinates.

Cylindrical Coordinate System

This coordinate system is used for problems with circular symmetry and involves transforming the Cartesian coordinates (x, y, z) to cylindrical coordinates (r, ?, z) using the relationships x = r cos ? and y = r sin ?. It simplifies the analysis of regions and functions that involve expressions like x² + y², which becomes r² in cylindrical coordinates.

Conversion of Regions in Cylindrical Coordinates

When converting a region defined in Cartesian coordinates to cylindrical coordinates, the inequalities that describe the region are rewritten in terms of r, ?, and z. For example, an expression like 0 ? x² + y² ? 4 converts to 0 ? r² ? 4 (or 0 ? r ? 2), and conditions such as y ? 0 translate into restrictions on the angular variable ?.

Transcript

00:01 We are given that e is the region in r3, such that 0 is less than equal to x squared plus y squared, which is less than equal to 4.

00:17 We have the y is greater than are equal to 0, and that z is between 0 and the plane 3 minus x.

00:33 We are also told that f of x, y, z is given by x squared plus y squared.

00:40 So we are going to first write all this in terms of cylindrical coordinates, and then we're going to evaluate the integral of f over the region e.

00:50 So let's start with e.

00:53 The first thing we notice is that x squared plus y squared is just equal to r squared.

00:58 So if r squared is between zero and four, that means r is between zero and two.

01:06 Now, y greater than are equal to 0.

01:09 So if you look at the xy plane here, then that's this region up here.

01:15 That's y greater than equal zero.

01:16 And so you can see there that theta now varies from 0 to pi.

01:26 And then the last one, we have z is bounded above by 3 minus, and now we just replace x with our cosine theta.

01:38 And then if we want to rewrite f in terms of our insolentrical quarters, coordinates.

01:46 So x squared plus y squared is just so that's our for part a we get all everything in cylindrical coordinates.

01:58 For part b now we're going to evaluate the triple integral over e of f of x y z dv so we have 0 to pi for theta.

02:12 For r we have 0 to 2 and for z we have 0 to 3 minus our cosine theta of in this case we have r.

02:22 Times the r squared in cylindrical coordinates is just r to the third.

02:27 Dz, dr, d theta.

02:32 Now we can't break this apart yet because we can see that the integral for z depends on r and theta...

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