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 function $f$ in cylindrical coordinates. b. Convert the integral $\iiint_{B} f(x, y, z) d V \quad$ into cylindrical coordinates and evaluate it. $f(x, y, z)=x+y$ $E=\left\{(x, y, z) | 1 \leq x^{2}+y^{2}+z^{2} \leq 2, z \geq 0, y \geq 0\right\}$

In the following exercises, the function $f$ and region $E$ are given.
a. Express the region $E$ and function $f$ in cylindrical coordinates.
b. Convert the integral $\iiint_{B} f(x, y, z) d V \quad$ into cylindrical coordinates and evaluate it.
$f(x, y, z)=x+y$
$E=\left\{(x, y, z) | 1 \leq x^{2}+y^{2}+z^{2} \leq 2, z \geq 0, y \geq 0\right\}$

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional coordinate system that extends polar coordinates by adding a vertical z-component. In this system, a point in space is represented by (r, ?, z), where r is the distance from the z-axis, ? is the angle measured from a fixed reference direction in the xy-plane, and z is the height above the xy-plane. This coordinate system is particularly useful when dealing with regions or symmetry related to cylinders or circular boundaries.

Jacobian and Volume Element

When converting integrals from Cartesian coordinates to cylindrical coordinates, the differential volume element changes to account for the new coordinate system. Specifically, the Jacobian determinant in the transformation introduces an extra factor of r, so that the volume element becomes dV = r dr d? dz. This adjustment is crucial for correctly computing integrals over volumes defined in cylindrical coordinates.

Conversion of Scalar Functions

To convert a scalar function from Cartesian to cylindrical coordinates, one substitutes the expressions for x and y in terms of r and ?; specifically, x = r cos? and y = r sin?. This process transforms the function into a form that can be directly integrated using the cylindrical coordinate system, enforcing consistency with the integration limits and the differential volume element.

Integration Domain Transformation

When a region of integration is described in Cartesian coordinates by inequalities, these inequalities must be re-expressed in cylindrical coordinates. This involves rewriting Cartesian expressions like x² + y² in terms of r, and adjusting inequalities for z as well as angular ranges to fit the radial and angular variables. Correct determination of the new limits for r, ?, and z is essential for accurately setting up and evaluating the corresponding integral in cylindrical coordinates.

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