Question

Write the integral \(\iiint_{D} f(r, \theta, z) d V\) as an iterated integral where \(D=\\{(r, \theta, z): G(r, \theta) \leq z \leq H(r, \theta), g(\theta) \leq r \leq h(\theta)\) \(\alpha \leq \theta \leq \beta\\}\).

Short answer

Question: Set up the triple integral of the function f(r, θ, z) over the given region D in cylindrical coordinates with limits g(θ) ≤ r ≤ h(θ), α ≤ θ ≤ β, and G(r, θ) ≤ z ≤ H(r, θ) in the order of integration r, θ, z. Answer: The iterated integral for the function f(r, θ, z) over the region D in cylindrical coordinates is given by: $$\int_{\alpha}^{\beta} \left(\int_{g(\theta)}^{h(\theta)} \left(\int_{G(r,\theta)}^{H(r,\theta)} f(r, θ, z) dz \right) dr \right) d\theta$$

Step-by-step solution

Write down the integral with respect to z

For z, we are considering the limits \(G(r, \theta) \leq z \leq H(r, \theta)\). Thus, the first integral will be: $$\int_{G(r, \theta)}^{H(r, \theta)} f(r, \theta, z) dz$$

Write down the integral with respect to r

For r, we are considering the limits \(g(\theta) \leq r \leq h(\theta)\). Thus, we have the double integral: $$\int_{g(\theta)}^{h(\theta)} \left(\int_{G(r,\theta)}^{H(r,\theta)} f(r, \theta, z) dz \right) dr$$

Write down the integral with respect to θ

For θ, we are considering the limits \(\alpha \leq θ \leq \beta\). Thus, we now have the triple integral: $$\int_{\alpha}^{\beta} \left(\int_{g(\theta)}^{h(\theta)} \left(\int_{G(r,\theta)}^{H(r,\theta)} f(r, \theta, z) dz \right) dr \right) d\theta$$ This is our required iterated integral.

Key concepts

Triple Integral

Understanding the triple integral concept is essential when dealing with the calculation of volumes or mass in three-dimensional space. A triple integral allows us to integrate a function of three variables, typically represented as \( f(x, y, z) \), over a three-dimensional region. The integral combines three single integrals into one, which are evaluated in succession, each representing integration over one dimension.

The process of solving a triple integral involves breaking down a complex, multi-dimensional shape into simpler parts, similar to dissecting a 3D object into numerous tiny cubes, adding up their contributions, and considering the limits of integration for each variable that defines the region of interest. In the context of the textbook exercise, the triple integral takes the form \( f(r, \theta, z) \) over a region \( D \) defined by specific functions of \( r \) and \( \theta \) which determine the boundaries for \( z \) and \( r \) as well as fixed limits for \( \theta \) itself.

Spherical Coordinates

Spherical coordinates are a system for representing points in three-dimensional space using three coordinates: the radius \( r \), the polar angle \( \theta \), often referred to as \( \theta \) in the context of spherical coordinates, and the azimuthal angle \( \phi \). This system is particularly useful when dealing with problems that have spherical symmetry, such as gravitational fields, and when calculating integrals over spherical volumes.

Instead of \( x, y, z \), which are used in Cartesian coordinates, spherical coordinates assume a point is determined by how far away it is from the origin (radius \( r \)), its angle from the z-axis (polar angle \( \theta \) ), and its angle from the x-axis within the x-y plane (azimuthal angle \( \phi \) ). In mathematical notation, the switch from Cartesian to spherical coordinates is expressed by the set of equations: \( x = r \sin(\theta)\cos(\phi) \), \( y = r \sin(\theta)\sin(\phi) \), and \( z = r \cos(\theta) \).

Integration Limits

Integration limits define the boundaries over which we evaluate an integral. In single-variable calculus, these boundaries are simply two points on the number line for a definite integral. However, in higher dimensions, particularly when evaluating triple integrals, the limits can be functions that define the bounds of a more complex region.

For three-dimensional integration, the integration limits dictate the geometry of the volume beneath the surface represented by the function being integrated. These limits can depend on one another, as seen in the iterated integral for a triple integral, where the limits of \( z \) depend on both \( r \) and \( \theta \) (\( G(r, \theta) \leq z \leq H(r, \theta) \) ), and the limits of \( r \) depend on \( \theta \) (\( g(\theta) \leq r \leq h(\theta) \) ). The outermost limits for \( \theta \) remain constants (\( \alpha \leq \theta \leq \beta \) ). When setting up an iterated integral, correctly determining these limits is crucial for accurately evaluating the integral and, consequently, determining the desired quantity, such as volume or mass.