Question

Transform to cylindrical coordinates but do not evaluate: a) \(\iiint_{R_{x z}} x^{2} y d x d y d z\), where \(R_{x y z}\) is the region \(x^{2}+y^{2} \leq 1,0 \leq z \leq 1\); b) \(\int_{0}^{1} \int_{0}^{\sqrt{1-x^{2}}} \int_{0}^{1+x+y}\left(x^{2}-y^{2}\right) d z d y d x\).

Short answer

Question: Transform the following triple integrals into cylindrical coordinates: a) The region \(R_{xyz}\) defined by \(x^2+y^2 \leq 1\) and \(0 \leq z \leq 1\): Integral (a) in cylindrical coordinates is: $$ \iiint_{R_{r\theta z}} r^2 \cos^2\theta r \sin\theta r dz dr d\theta $$ with limits: \(0 \le r \le 1\), \(0 \le z \le 1\), and \(0 \le \theta \le 2\pi\). b) The triple integral given by: $$ \int_{0}^{1}\int_{0}^{\sqrt{1-x^2}} \int_{0}^{1+x+y}(x^2-y^2) dy dz dx $$ Integral (b) in cylindrical coordinates is: $$ \int_{0}^{1}\int_{0}^{\sqrt{1-r^2\cos^2\theta}} \int_{0}^{1+r\cos\theta + r\sin\theta}(r^2\cos^2\theta - r^2\sin^2\theta)r dz dr d\theta $$ with limits: \(0 \le r \le 1\), \(0 \le r\sin\theta \le \sqrt{1-r^2\cos^2\theta}\), and \(0 \le z \le 1+r\cos\theta + r\sin\theta\).

Step-by-step solution

Transform integral (a) into cylindrical coordinates

For integral (a), the relationship between rectangular coordinates \((x,y,z)\) and cylindrical coordinates \((r,\theta,z)\) are given by: $$ x = r \cos\theta \\ y = r \sin\theta \\ z = z $$ Using these relationships, we can rewrite the triple integral (a) in cylindrical coordinates. To do that, we also need to take into account the Jacobian for the transformation from rectangular to cylindrical coordinates, which is \(r\). The new limits for \(r\) will be from \(0\) to \(1\), limits for \(\theta\) would be from \(0\) to \(2\pi\), and limits for \(z\) will remain from \(0\) to \(1\). Now integral (a) becomes: $$ \iiint_{R_{r\theta z}} r^2 \cos^2\theta r \sin\theta r dz dr d\theta $$ with limits: \(0 \le r \le 1\), \(0 \le z \le 1\), and \(0 \le \theta \le 2\pi\).

Transform integral (b) into cylindrical coordinates

For integral (b), we first rewrite the given limits in terms of \(r\) and \(\theta\). The limits for \(x\) are given by: \(0 \le x \le 1\) which translates to \(0 \le r\cos\theta \le 1\) in cylindrical coordinates. The limits for \(y\) are given by: \(0 \le y \le \sqrt{1-x^2}\) which becomes \(0 \le r\sin\theta \le \sqrt{1-r^2\cos^2\theta}\) in cylindrical coordinates. The limits for \(z\) are given by: \(0 \le z \le 1+x+y\) which becomes \(0 \le z \le 1+r\cos\theta + r\sin\theta\) in cylindrical coordinates. Finally, we rewrite the integrand \((x^2-y^2)\) in cylindrical coordinates which become \((r^2\cos^2\theta - r^2\sin^2\theta)\). Now, integral (b) becomes: $$ \int_{0}^{1}\int_{0}^{\sqrt{1-r^2\cos^2\theta}} \int_{0}^{1+r\cos\theta + r\sin\theta}(r^2\cos^2\theta - r^2\sin^2\theta)r dz dr d\theta $$ with limits: \(0 \le r \le 1\), \(0 \le r\sin\theta \le \sqrt{1-r^2\cos^2\theta}\), and \(0 \le z \le 1+r\cos\theta + r\sin\theta\) in cylindrical coordinates.

Key concepts

Triple Integral

The concept of a triple integral extends the idea of integrating functions of two variables to functions of three variables. A triple integral, denoted as \(\iiint\), calculates the volume under the surface defined by the function in a three-dimensional space. It involves three successive integrations, each corresponding to one of the three variables that define the spatial domain. In the context of cylindrical coordinates, a triple integral can be used to calculate physical quantities such as mass, charge, and volume for objects with cylindrical symmetry.

To set up a triple integral in cylindrical coordinates, one must consider the symmetry of the problem and adjust the limits of integration for \(r\), the radial coordinate, \(\theta\), the angular coordinate, and \(z\), the height coordinate. This approach simplifies the integration process when dealing with problems involving circular cross-sections or rotational symmetry about the z-axis.

Coordinate Transformation

Coordinate transformation is the process of changing the perspective of a given space from one coordinate system to another. This method is essential when the geometry of the space or the symmetry of the problem makes it more convenient to use a particular coordinate system. A common scenario where such a transformation is applied is when dealing with problems involving cylindrical shapes or rotational symmetry.

In the process of transforming from rectangular \( (x,y,z) \) to cylindrical \( (r,\theta,z) \) coordinates, we use the following relationships: \(
x = r \cos\theta,
y = r \sin\theta,
z = z
\) Here, \(r\) and \(\theta\) represent the radial distance from the origin and the angle in the xy-plane from the positive x-axis, respectively. The variable \(z\) remains the same in both coordinate systems.

Jacobian Determinant

The Jacobian determinant is a function that describes how much the volume element changes when transforming from one coordinate system to another. It is a critical component of the integration process, as it compensates for the distortion that happens during the coordinate transformation.

When converting from rectangular to cylindrical coordinates, the area element \(dx dy\) in the xy-plane transforms into a new area element \(r dr d\theta\). The factor \(r\) is the Jacobian determinant for this transformation, indicating that the area stretches by a factor proportional to the distance from the origin as one moves away from the axis of symmetry. This factor must be included when rewriting the integrand expressions in the new cylindrical coordinates to ensure that the integral's value remains accurate.

Rectangular to Cylindrical Conversion

Converting functions and limit bounds from rectangular to cylindrical coordinates is a process that involves substituting the rectangular expressions of \(x\), \(y\), and \(z\) with their cylindrical equivalents. It streamlines the evaluation of triple integrals for regions with rotational symmetry.

As part of the transformation process, every occurrence of \(x\) and \(y\) in the integrand and the limits must be replaced with \(r\cos\theta\) and \(r\sin\theta\), respectively. The resulting integral will be expressed entirely in terms of \(r\), \(\theta\), and \(z\), often simplifying the integrand and the evaluation of bounds. This conversion enables the straightforward application of cylindrical symmetry in solving the integral, which is evident from the limits and integrand of the given example exercises.