Problem 14 - Thomas Calculus

Question

Convert the integral $$ \int_{-1}^{1} \int_{0}^{\sqrt{1-y^{2}}} \int_{0}^{x}\left(x^{2}+y^{2}\right) d z d x d y $$ to an equivalent integral in cylindrical coordinates and evaluate the result.

Step-by-Step Solution

Answer

The integral evaluates to 0.

Step 1: Understand the Geometry

The limits of integration suggest the region of integration is bounded by \[0 \leq z \leq x, \quad 0 \leq x \leq \sqrt{1-y^2}, \quad -1 \leq y \leq 1.\] This implies the region is part of a cylindrical section with radius 1. It forms a segment of a half-cylinder in the x-y plane with varying height along x.

Step 2: Define Cylindrical Coordinates

In cylindrical coordinates, x is replaced by \( r \cos(\theta) \), y by \( r \sin(\theta) \), and the Jacobian of transformation is \( r \). Thus, the volume element transforms as \[ dV = r \, dr \, d\theta \, dz. \]

Step 3: Express Limits in Cylindrical Coordinates

- The cylindrical radius \( r \) ranges from 0 to 1 (since \( \sqrt{x^2 + y^2} = 1 \)).- \( \theta \) sweeps half a circle, from 0 to \( \pi \).- The vertical height z ranges from 0 to \( r \cos(\theta) \) (since \( z \leq x \)).Thus, the new limits become:\[ \int_{0}^{\pi} \int_{0}^{1} \int_{0}^{r\cos\theta} \ldots \, dz \, dr \, d\theta \] after accounting for the region and substitutions.

Step 4: Convert the Integral

Convert the given integral using cylindrical coordinates:\[ \int_{0}^{\pi} \int_{0}^{1} \int_{0}^{r \cos \theta} (r^2) \, dz \, dr \, d\theta, \]since \( x^2 + y^2 = r^2 \).

Step 5: Evaluate the Innermost Integral

Evaluate \( \int_{0}^{r\cos\theta} r^2 \, dz \). - The integral evaluates to \[ r^2 z \bigg|_{0}^{r \cos \theta} = r^3 \cos \theta. \]

Step 6: Evaluate the Middle Integral

Substituting the result of the innermost integral, evaluate \( \int_{0}^{1} r^3 \cos \theta \, dr \).- This evaluates to \[ \frac{r^4}{4} \cos \theta \bigg|_{0}^{1} = \frac{1}{4} \cos \theta. \]

Step 7: Evaluate the Outer Integral

Evaluate the result of the previous integral over \( \theta \), \[ \int_{0}^{\pi} \frac{1}{4} \cos \theta \, d\theta. \]- This integral evaluates as follows:\[ \frac{1}{4} \sin \theta \bigg|_{0}^{\pi} = 0. \]

Step 8: Conclusion

The original integral evaluates to 0 in the cylindrical coordinate system, which implies the symmetry cancels out contributions over this specific region.

Key Concepts

Triple Integrals

Triple integrals are a powerful mathematical tool used for calculating volumes and other quantities in three-dimensional space. They extend the concept of integration from one or two dimensions into the third, allowing us to handle more complex problems.

When evaluating a triple integral, we integrate a function with respect to three variables, usually corresponding to spatial dimensions. The notation feels like a step up from double integrals, incorporating a third layer of complexity. For example, given a function \(f(x, y, z)\), the triple integral over a cuboidal region \(R\) is expressed as \[\int \int \int_R f(x, y, z) \, dV,\]where \(dV\), the volume element, is written as \(dz \, dy \, dx\) and indicates the accumulation of infinitesimal volumes. Triple integrals are not confined to cuboidal regions and can apply to more intriguing shapes by appropriately adjusting the integration limits.

They become particularly exciting when we transition to another coordinate system, such as cylindrical or spherical, hence adapting the volume element to the symmetry present in the problem. This adaptation is crucial for efficiently solving integrals over non-standard regions.

Coordinate Transformation

Coordinate transformations are incredibly useful in calculus, particularly when dealing with triple integrals. They allow us to convert a problem expressed in one coordinate system into another, which might be more suited to the problem's geometry.

In the context of cylindrical coordinates, we transition from Cartesian coordinates \((x, y, z)\) to cylindrical \((r, \theta, z)\). This is particularly advantageous when dealing with objects or regions exhibiting cylindrical symmetry. The conversion formulas are:

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

The reverse is also true if needed, but typically we use the forward transformation to ease the integration over cylindrical regions.

Transforming coordinates can simplify integration limits and equations, and sometimes reveal symmetries that reduce computational effort. It's essential for students to appreciate that selecting an optimal coordinate system based on the geometry of the problem is a heuristic approach that can greatly simplify the solution.

Integration Limits

Integration limits dictate where the integration over a specified region starts and ends. In the world of triple integrals, they define the bounds of each integrated variable and shape the region of interest.

When switching to cylindrical coordinates, defining these limits properly becomes critical. This step ensures you are integrating over the correct region. For instance, if integrating within a cylinder, the radius \(r\) might vary from 0 to the cylinder's maximum radius, the angle \(\theta\) could range from 0 to \(2\pi\), depending on the extent of the circle covered, and the height \(z\) will have bounds that correspond to the cylinder's top and bottom planes.

• For our problem, \(r\) ranges from 0 to 1.
• The angle \(\theta\) goes from 0 to \(\pi\), covering half a circle.
• Finally, \(z\) varies between 0 and \(r \cos(\theta)\), conforming to the original geometry given by the bounds \(z \leq x\).

Defining integration limits properly not only ensures accuracy but also sets the stage for effective computational approaches, especially when evaluating integrals analytically.

Jacobian Determinant

The Jacobian determinant is a key part of coordinate transformations in calculus. When changing variables during integration, the Jacobian accounts for how the transformation affects the volume elements of space.

For moving from Cartesian to cylindrical coordinates, the Jacobian serves to adjust the integration measure (or volume element) appropriately. This adjustment reflects the way areas and volumes stretch under the transformation. For cylindrical coordinates, the Jacobian determinant is \(r\), transforming the volume element from \(dx \, dy \, dz\) to \(r \, dr \, d\theta \, dz\).

It is crucial to include the Jacobian when converting integrals from one coordinate system to another. Omitting it results in incorrect calculations because the Jacobian scales the differential volume element to compensate for the differing shapes and sizes of the regions in each coordinate system.

• The Jacobian for Cartesian to cylindrical is simple yet essential: it makes sure that the transition respects the geometry of space in which the function is evaluated.

In practice, multiply the integrand by the Jacobian during substitution to ensure that integration accounts for how space is warped or stretched, thus maintaining accuracy and preserving the volume calculated under the new coordinates.