91Ó°ÊÓ

A triple integral allows us to calculate the volume under a surface in three-dimensional space. It extends the concept of a double integral to three dimensions. In the given exercise, we deal with the function \(f(x, y, z) = y\) over the region \(E\). The goal is to express and evaluate this triple integral using cylindrical coordinates. The process involves integrating the function \(f(r, \theta, y)\) over the volume \(E\). Triple integrals typically consist of three nested integrals, one for each coordinate: \(r\) (radial distance), \(\theta\) (angle), and \(y\) (height). Each integral evaluates the accumulation of the function over a specific interval of the coordinate.Using these integrals effectively requires understanding the specific boundaries and transformations applied to the region \(E\), especially when converting from Cartesian to cylindrical coordinates.

Cylindrical coordinates are a three-dimensional coordinate system where any point can be expressed using a radius \(r\), an angle \(\theta\), and a height \(y\). These are particularly helpful when dealing with problems involving symmetrical shapes around an axis, like cylinders. To convert Cartesian coordinates \((x, y, z)\) into cylindrical coordinates, we use:

• \(x = r\cos\theta\)
• \(z = r\sin\theta\)
• \(y = y\)

In the exercise, the given condition \(1 \leq x^2 + z^2 \leq 9\) is transformed into \(1 \leq r^2 \leq 9\), simplifying to \(1 \leq r \leq 3\). This conversion reformulates the problem, making it easier to integrate due to the symmetry around the z-axis.

Volume integration refers to integrating a function over a three-dimensional space. With cylindrical coordinates, the volume element \(dV\) is expressed as \(r \, dr \, d\theta \, dy\). This expression accounts for the radial nature of cylindrical coordinates and is crucial to setting up the triple integral.The exercise demonstrates setting up a triple integral in cylindrical coordinates, where the bounds for each integral represent the limits of the region \(E\) in terms of \(r\), \(y\), and \(\theta\). The triple integral in the exercise calculates the accumulated value of the function \(f = y\) over the specified region, representing a volume. Performing this computation requires evaluating each integral one by one, starting from the innermost to the outermost.

Defining the boundaries is crucial in setting up integrals correctly. In this exercise, we translate the initial conditions to cylindrical form:

• The radial coordinate \(r\) is bounded by the constraints given by \(1 \leq r \leq 3\).
• The \(y\) coordinate is constrained by the inequality \(0 \leq y \leq 1-r^2\), interpreted as the height boundary for each fixed \(r, \theta\) pair.
• The angle \(\theta\) varies freely from \(0\) to \(2\pi\), covering a full rotation around the z-axis.

Aligning the boundaries with the cylindrical framework ensures that when the triple integral is set up, it accurately reflects the volume of interest. This alignment is an essential step before proceeding to evaluate the integral.