Problem 20 - Thomas Calculus

Question

Set up the iterated integral for evaluating \(\iiint_{D} f(r, \theta, z) d z r d r d \theta\) over the given region \(D .\) \(D\) is the prism whose base is the triangle in the \(x y\) -plane bounded by the \(y\) -axis and the lines \(y=x\) and \(y=1\) and whose top lies in the plane \(z=2-x\)

Step-by-Step Solution

Answer

The integral is \( \int_0^{\pi/4} \int_0^1 \int_0^{2-r\cos(\theta)} f(r, \theta, z) \, dz \, r \, dr \, d\theta \).

Step 1: Determine the boundaries in the xy-plane

The base of the prism is a region in the xy-plane defined by the y-axis and the lines y=x and y=1. This means the base is a right triangle with vertices at (0,0), (1,1), and (0,1). The line y=x starts from (0,0) and goes through (1,1), and the horizontal line y=1 intersects the y-axis at (0,1). Thus, y varies from 0 to 1 and for a fixed y, x varies from y to 1.

Step 2: Determine the boundaries for z

The top of the prism is given by the plane z=2-x. Since x varies between y and 1 based on our earlier setup, the z-coordinate will vary from 0 (at the base) to 2-x at the top. Hence, for a given point (x, y), z varies from 0 to 2-x.

Step 3: Express the region D in cylindrical coordinates

Convert to cylindrical coordinates, where x = r*cos(θ), y = r*sin(θ), and we already have z. In this problem, the triangle in the xy-plane corresponds to 0 ≤ θ ≤ π/4 (since y = x implies θ = π/4) and 0 ≤ r ≤ 1, reflecting the dimensions of the triangle within the unit circle.

Step 4: Setup of the iterated integral

Using the determined bounds, the iterated integral in cylindrical coordinates is set up as follows:\[\int_0^{π/4} \int_0^1 \int_0^{2-r\cos(θ)} f(r, θ, z) \, dz \, r \, dr \, dθ\]Here, the order of integration is dz, dr, and dθ in line with the problem statement.

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates are a way of representing points in a three-dimensional space by using a combination of polar coordinates in the xy-plane, along with a z-coordinate. They are particularly useful when dealing with problems involving symmetry around an axis, such as cylindrical shapes or when integrating over regions that are simpler in polar than in Cartesian coordinates.

In cylindrical coordinates, the position of a point is described by three values:

• \(r\): the distance from the point to the z-axis (analogous to the radius in polar coordinates)
• \(\theta\): the angle between the positive x-axis and the line connecting the point’s projection in the xy-plane and the origin
• \(z\): the height above the xy-plane, same as in Cartesian coordinates

The conversion formulas between Cartesian coordinates \((x, y, z)\) and cylindrical coordinates \((r, \theta, z)\) are:

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

Using these conversions, regions that may appear complex in Cartesian coordinates can often be more easily described and integrated in cylindrical coordinates.

Iterated Integrals

Iterated integrals are a method used to evaluate multiple integrals by breaking them down into a series of single integrals. This process involves integrating a function one variable at a time in a set order, making the solution more approachable and computationally feasible.

The sequence of operations generally follows the order of the variables specified in the integral. For example, if you have an iterated integral like \(\int \int_{D} f(x, y) \, dx \, dy\), it suggests integrating first with respect to \(x\) and then with respect to \(y\). The boundaries for each integration can depend on the preceding variable, ensuring that the final volume or area corresponds accurately to the region described.

Iterated integrals in cylindrical coordinates take advantage of the simplification provided by the coordinate transformation, especially when the geometry naturally matches cylindrical symmetry.

Triple Integrals

Triple integrals extend the idea of integrating over an area to integrating over a three-dimensional volume. They are crucial in calculating volumetric properties like mass or charge distributions over a space. In terms of an integral, this involves adding an additional layer, going from double integrals (areas) to triple integrals (volumes).

In cylindrical coordinates, a triple integral can be expressed in the form:\[\iiint_{V} f(r, \theta, z) \, dz \, r \, dr \, d\theta\]Here, note the additional \(r\) factor, which arises from the Jacobian determinant when converting from Cartesian to cylindrical coordinates. This factor adjusts the volume element to correctly represent the cylindrical geometry.

Triple integrals in cylindrical coordinates help streamline the integration process when the function or boundaries exhibit cylindrical or rotational symmetry. This method involves satisfying the limits for \(r\), \(\theta\), and \(z\), ensuring a thorough evaluation across the described region.