Problem 36 - 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. \(\iiint_{D} f(r, \theta, z) d z r d r d \theta\) over the given region \(D\)

Step-by-Step Solution

Answer

The evaluated integral in cylindrical coordinates is \(\frac{\pi}{5}\).

Step 1: Understand the Region

The given integral is in Cartesian coordinates where \(x\) ranges from \(0\) to \(\sqrt{1-y^2}\), \(y\) ranges from \(-1\) to \(1\), and \(z\) ranges from \(0\) to \(x\). This region corresponds to a quarter of a circle when rotated around the y-axis.

Step 2: Convert Limits to Cylindrical Coordinates

In cylindrical coordinates, \(x^2 + y^2 = r^2\). Thus, the circular region from \(-1\) to \(1\) for \(y\) and \(0\) to \(\sqrt{1-y^2}\) for \(x\) is converted to \(r = 0\) to \(1\) and \(\theta = 0\) to \(\pi\). The \(z\) range remains from \(0\) to \(r\).

Step 3: Convert Function and Evaluate Jacobian

The function \(x^2 + y^2\) becomes \(r^2\) in cylindrical coordinates. The differential \(dx\, dy\, dz\) becomes \(r \, dr \, d\theta \, dz\) due to the cylindrical coordinate conversion.

Step 4: Set up the New Integral

The integral in cylindrical coordinates becomes: \[\int_{0}^{\pi} \int_{0}^{1} \int_{0}^{r} r^2 \cdot r \, dz \, dr \, d\theta\]Which simplifies to \[\int_{0}^{\pi} \int_{0}^{1} \int_{0}^{r} r^3 \, dz \, dr \, d\theta\]

Step 5: Evaluate the Integral with respect to \'z\'

First integrate with respect to \(z\): \[\int_{0}^{r} r^3 \, dz = [r^3z]_{0}^{r} = r^4\] Thus, the integral becomes: \[\int_{0}^{\pi} \int_{0}^{1} r^4 \, dr \, d\theta\]

Step 6: Evaluate the Integral with respect to \'r\'

Now, integrate with respect to \(r\): \[\int_{0}^{1} r^4 \, dr = \left[ \frac{r^5}{5} \right]_{0}^{1} = \frac{1}{5}\]This reduces the integral to: \[\int_{0}^{\pi} \frac{1}{5} \, d\theta\]

Step 7: Evaluate the Integral with respect to \'\theta\'

Finally, integrate with respect to \(\theta\): \[\int_{0}^{\pi} \frac{1}{5} \, d\theta = \left[ \frac{\theta}{5} \right]_{0}^{\pi} = \frac{\pi}{5}\]

Step 8: Final Result

The result of the integral in cylindrical coordinates is \(\frac{\pi}{5}\).

Key Concepts

Triple Integrals

Triple integrals extend the concept of integration to three-dimensional space, allowing us to calculate volumes and other quantities in 3D. In essence, a triple integral operates much like a double integral, but it considers one additional dimension.
Imagine wanting to find the volume of a complex 3D shape. A triple integral allows you to break this shape into infinitesimally small volume elements, sum them up, and determine the total volume. This technique can also be applied to integrals of functions over a volume, such as calculating mass or charge distributions.

• The innermost integral simplifies first, addressing one variable at a time.
• The limits of integration define the region of space to be considered for the computation.
• In our exercise, the triple integral is used to determine the volume of a region bounded in space.

By converting the triple integral into cylindrical coordinates, evaluating such regions becomes significantly more manageable and intuitive, especially when the region has a circular symmetry.

Coordinate Transformation

Switching from Cartesian to cylindrical coordinates is a crucial step in handling certain integrals, particularly those involving circular symmetry.
In cylindrical coordinates, a point in space is represented by a radius, an angle, and a height, denoted as \(r, \theta, z\).
These coordinates simplify the integration process for problems involving cylinders or regions symmetric about an axis. Conversion formulas for such transformations are:

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

In our exercise, the transformation took advantage of radial symmetry. The change of variables also involves using the Jacobian determinant, given by \( r \) when switching from Cartesian to cylindrical coordinates.
This multiplier accounts for the "stretching" of space as you move to polar dimensions. It is essential for converting area elements correctly when performing integration.

Volume Integration

Volume integration is a powerful technique to compute the size or volume of a solid in space. This is particularly relevant in physics and engineering, where understanding the distribution of mass, density, or other variables within a volume is key.Converting the integral into cylindrical coordinates was specifically beneficial because the original region described a cylinder segment.
With cylindrical integration:

• The radial limits \( r = 0 \) to \( r = 1 \) concern the distance from the origin to the edge of the circle.
• The angular limits \( \theta = 0 \) to \( \pi \) cover half the circle, ensuring only the specified region is calculated.
• The height limits \( z = 0 \) to \( z = r \) reflect the constraints given by the initial Cartesian parameters.

The advantage of utilizing volume integration via triple integrals is the ability to set exact boundaries for each axis, ensuring accurate computation of the specific volume of interest. This approach succinctly encapsulates a method to transform and solve integrals concerning complex 3D shapes.

Calculus Exercises

Through calculus exercises like the one discussed, students learn essential skills for solving real-world problems. They present opportunities to apply theoretical concepts like triple integrals and coordinate transformation practically. Engaging with such exercises helps to:

• Enhance spatial reasoning by translating geometric regions into mathematical models.
• Develop a comprehensive understanding of integrating functions over volumes.
• Foster the ability to manipulate mathematical expressions through variable change.

Breaking down a calculus problem into manageable parts, as shown in the step-by-step solution, empowers learners to tackle complex problems systematically. Each exercise reinforces core concepts while guiding students on when and how to apply specific calculus strategies, such as switching coordinate systems. These are critical skills for pursuing more advanced studies or professions in STEM fields.