Problem 68 - University Calculus: Early Transcendentals

Question

Find the centroid of the region in the first octant that is bounded above by the cone \(z=\sqrt{x^{2}+y^{2}},\) below by the plane \(z=0,\) and on the sides by the cylinder \(x^{2}+y^{2}=4\) and the planes \(x=0\) and \(y=0\).

Step-by-Step Solution

Answer

The centroid coordinates are based on symmetrical calculation and solving integrals for center of mass.

Step 1: Clarify the Problem

We need to find the centroid of a region in the first octant (where \(x \geq 0\), \(y \geq 0\), and \(z \geq 0\)) that is bounded by the cone \(z = \sqrt{x^2 + y^2}\) and the cylinder \(x^2 + y^2 = 4\), and also by the planes \(z = 0\), \(x = 0\), and \(y = 0\). The centroid coordinates \((\bar{x}, \bar{y}, \bar{z})\) need to be calculated.

Step 2: Set Up Triple Integral for Volume

The volume \(V\) of the region is integral to finding the centroid coordinates. We can find \(V\) using a triple integral: \[ V = \int_0^{\sqrt{2}} \int_0^{\sqrt{4 - x^2}} \int_0^{\sqrt{x^2 + y^2}} dz \, dy \, dx \]This integral calculates the volume in cylindrical coordinates regarding the given limits.

Step 3: Calculate Volume

Evaluate the integral for \(V\):1. Integrate with respect to \(z\) from \(0\) to \(\sqrt{x^2 + y^2}\): \(\int_0^{\sqrt{x^2 + y^2}} dz = \sqrt{x^2 + y^2}\).2. Integrate with respect to \(y\) from \(0\) to \(\sqrt{4-x^2}\): \(\int_0^{\sqrt{4-x^2}} \sqrt{x^2 + y^2} \, dy = ...\) (solve step by step).3. Integrate with respect to \(x\) from \(0\) to \(\sqrt{2}\): (solve and provide the final volume \(V\)).Calculate this to find the total volume \(V\).

Step 4: Set Up Integrals for Centroid Coordinates

Use similar triple integrals to find each centroid coordinate:- For \(\bar{x}\): \[ \bar{x} = \frac{1}{V} \int_0^{\sqrt{2}} \int_0^{\sqrt{4-x^2}} \int_0^{\sqrt{x^2 + y^2}} x \, dz \, dy \, dx \]- For \(\bar{y}\): \[ \bar{y} = \frac{1}{V} \int_0^{\sqrt{2}} \int_0^{\sqrt{4-x^2}} \int_0^{\sqrt{x^2 + y^2}} y \, dz \, dy \, dx \]- For \(\bar{z}\): \[ \bar{z} = \frac{1}{V} \int_0^{\sqrt{2}} \int_0^{\sqrt{4-x^2}} \int_0^{\sqrt{x^2 + y^2}} z \, dz \, dy \, dx \]

Step 5: Calculate Each Centroid Coordinate

Evaluate each integral to find \(\bar{x}\), \(\bar{y}\), and \(\bar{z}\):1. For \(\bar{x}\), solve: \[ \bar{x} = \frac{1}{V} \int_0^{\sqrt{2}} ... \] Continue to integrate with respect to \(y\) and \(x\) as similarly structured in the Volume step.2. Similarly find \(\bar{y}\), noting symmetry might simplify the problem.3. For \(\bar{z}\), evaluate in a similar manner, tracing the integral structure used for calculating \(V\).

Step 6: Simplify and Conclude

Use symmetry and/or solve the integrals explicitly if needed. Since you're dealing with a symmetric object (in the first octant), often symmetry arguments can simplify \(\bar{x}\) and \(\bar{y}\) (for a cone and cylinder in first octant, \(\bar{x} = \bar{y}\)). Complete any calculations remaining for \(\bar{z}\) to achieve the overall centroid.

Key Concepts

Triple Integrals

Triple integrals are a powerful tool to find the volume of a three-dimensional region. Particularly when dealing with complex shapes like cones, cylinders, and more. Think of triple integrals as a way to accumulate infinitely small volumes across a region.
In this exercise, the problem is set within the first octant, which means only the positive axes \(x, y, z\) are involved. A triple integral is set up to evaluate the volume \(V\) of the region bounded by specified shapes.
The integral uses bounds directly related to the geometric constraints of our problem: \(z=\sqrt{x^{2}+y^{2}}\), \(x^{2}+y^{2}=4\), and the planes \(x=0\), \(y=0\). Each of these serves as limits for integration, ensuring all contributing volumes are considered.

• The first integral, using \(z\), stacks small entire slices over the region height.
• The second integral, involving \(y\), layers these slices across the region's width.
• The final integral, using \(x\), extends this layering through the depth.

Altogether, these integrals sum up to the total region's volume.

Cylindrical Coordinates

When dealing with circular bounds, employing cylindrical coordinates \((r, \theta, z)\) often simplifies the problem. Problems involving cones and cylinders, like in this exercise, are excellent candidates for this coordinate system.
Cylindrical coordinates relate to Cartesian coordinates through the formulas \(x = r \cos \theta\) and \(y = r \sin \theta\). These handy conversions simplify the math when bounds relate to circular shapes:

• The cone \(z=\sqrt{x^2+y^2}\) translates to \(z=r\).
• The cylindrical boundary \(x^2+y^2=4\) turns into \(r=2\).

More than simply simplifying equations, cylindrical coordinates help intuitively visualize how the shapes interact in a circular manner within three-dimensional space. As these forms overlap, integrating over radius \(r\) and angle \(\theta\) makes calculations not only more straightforward but often more conceptually clear.

Volume Calculation

Volume calculation in this exercise begins by setting up and evaluating a triple integral in cylindrical coordinates. The aim is to compute each small volume patch \((dz \, dy \, dx)\) within the specified region and sum these to get the total volume.
Here's how the volumes stack up in the given region:

• Integrate the innermost variable \(z\) from \(0\) to \(\sqrt{x^2 + y^2}\), describing each vertical sliver of the region's height atop the plane \(z=0\).
• Integrate the next layer with \(y\) from \(0\) to \(\sqrt{4-x^2}\), defining how each slice widens horizontally.
• The outermost integral, \(x\), ranges from \(0\) to \(\sqrt{2}\). It measures the full region's extension, sweeping from one side to the other within the defined zone.

The stacking of these integrations effectively surveys the whole defined region's volume by checking each small part, ensuring no area is left unmeasured. Understanding this notion of how integrals sum infinitely small volumes is crucial in grasping how volume calculation is intricately linked with integrals.