Evaluate Integral Using Polar Coordinates ∬D(x²+y²)⁹/² DA

Hey guys! Today, we're diving into a fun math problem that involves evaluating a double integral using polar coordinates. This technique is super useful when dealing with regions that have circular symmetry, like our disk D in this case. Let's break it down step by step so you can ace similar problems in the future!

Understanding the Problem

So, what are we trying to do? We have this integral: $\iint_D(x^2+y2)^{9/2} dA$, and D is the disk defined by $x^2+y2 \leq 4$. Essentially, we need to find the value of this integral over the circular region. The function we're integrating is $(x^2+y2)^{9/2}$, which looks a bit intimidating in Cartesian coordinates (x, y). But don't worry, polar coordinates are here to save the day!

Why Polar Coordinates?

You might be wondering, why bother with polar coordinates? Well, the key here is the shape of our region D – it's a disk! Polar coordinates (r, θ) are a natural fit for circles and disks because they describe points in terms of their distance from the origin (r) and the angle they make with the positive x-axis (θ). This makes the limits of integration much simpler to express and often simplifies the integrand itself.

Setting Up the Conversion

Before we jump into the integral, let's refresh our memory on how to convert between Cartesian and polar coordinates:

• x = r cos θ
• y = r sin θ

• $$x^2 + y^2 = r^2$$

• dA = r dr dθ (This is crucial! The area element dA transforms to r dr dθ in polar coordinates)

Now, let's apply this to our problem. Notice that $x^2 + y^2$ appears in our integrand. Using the conversion, we can replace it with $r^2$. This will simplify our expression significantly. Also, the inequality $x^2 + y^2 \leq 4$ tells us that the radius r of our disk ranges from 0 to 2 (since $r^2 \leq 4$). The angle θ covers the entire circle, so it ranges from 0 to 2π.

Transforming the Integral

Okay, we've got our conversions and our limits of integration. Now, let's rewrite the integral in polar coordinates. Our original integral is: $\iint_D(x^2+y2)^{9/2} dA$. Substituting $x^2 + y^2 = r^2$ and $dA = r dr dθ$, we get:

$$\iint_D(r^2)^{9/2} r dr dθ$$

Simplifying the integrand, we have:

$$\iint_D r^9 * r dr dθ = \iint_D r^{10} dr dθ$$

Now, we need to set up the limits of integration. As we discussed earlier, r goes from 0 to 2, and θ goes from 0 to 2π. So, our integral becomes:

$$\int_{0}^{2π} \int_{0}^{2} r^{10} dr dθ$$

Evaluating the Integral

Alright, the hard part is done! Now we just need to evaluate this double integral. We'll start with the inner integral with respect to r:

$$\int_{0}^{2} r^{10} dr = [\frac{r^{11}}{11}]_{0}^{2} = \frac{2^{11}}{11} - \frac{0^{11}}{11} = \frac{2048}{11}$$

Great! Now we have a constant value. Let's plug that into the outer integral with respect to θ:

$$\int_{0}^{2π} \frac{2048}{11} dθ = \frac{2048}{11} [θ]_{0}^{2π} = \frac{2048}{11} (2π - 0) = \frac{4096π}{11}$$

The Final Answer

So, after all that, we've found that the value of the integral $\iint_D(x^2+y2)^{9/2} dA$ over the disk D is $\frac{4096π}{11}$. Not too shabby, right?

Key Takeaways

• Polar coordinates are your best friend when dealing with circular regions. They simplify the limits of integration and often the integrand itself.
• Remember the Jacobian! The area element transforms to $dA = r dr dθ$ in polar coordinates. Forgetting this is a common mistake.
• Practice makes perfect. The more you work with polar coordinates, the more comfortable you'll become with them.

Real-World Applications of Polar Coordinates and Double Integrals

Understanding polar coordinates and double integrals isn't just about acing math exams; it has real-world applications in various fields. Let's explore some exciting examples:

Physics and Engineering

In physics, polar coordinates are essential for describing circular motion, wave phenomena, and electromagnetic fields. Imagine calculating the electric field generated by a charged disk – double integrals in polar coordinates come to the rescue! Engineers use these concepts to design antennas, analyze fluid flow in circular pipes, and model heat transfer in cylindrical objects. For instance, when analyzing the heat distribution across a circular plate, engineers use double integrals in polar coordinates to determine the temperature at various points. This is crucial for designing efficient cooling systems or ensuring the structural integrity of components exposed to heat.

Image Processing

Ever wondered how images are rotated or manipulated in software like Photoshop? Polar coordinates play a key role in image processing. Transforming an image to polar coordinates can simplify tasks like detecting circular objects or applying radial filters. For example, if you want to blur an image radially (making it blurry further away from the center), working in polar coordinates makes the math much easier. Double integrals are used here to calculate properties like the average pixel intensity over a circular region, which can be helpful for image segmentation or object recognition. Imagine trying to identify circular features in a noisy image – converting to polar coordinates and applying filtering techniques simplifies the process significantly.

Probability and Statistics

Believe it or not, polar coordinates even show up in probability and statistics! The normal distribution, a cornerstone of statistics, has a probability density function that involves a Gaussian function (which has a circular symmetry). When dealing with two-dimensional normal distributions, polar coordinates can simplify calculations of probabilities and expectations. Double integrals are used to calculate probabilities over regions in the plane, and using polar coordinates can make these calculations more tractable, especially for circular regions or distributions with circular symmetry. For example, consider analyzing the spread of data points around a central point – polar coordinates can provide valuable insights into the distribution's shape and characteristics.

Computer Graphics and Game Development

Creating realistic and visually appealing graphics often involves working with circles, arcs, and other curved shapes. Polar coordinates are invaluable in computer graphics for generating these shapes, defining lighting effects, and creating textures. For instance, think about rendering a circular spotlight or simulating the reflection of light off a spherical object – polar coordinates help define the geometry and perform the necessary calculations. Double integrals might be used to calculate the total light intensity falling on a surface, contributing to the realism of the rendering. In game development, polar coordinates are used to control the movement of characters or projectiles in circular paths, create visual effects like swirling patterns, and design game environments with circular layouts.

Geospatial Analysis

Geospatial analysis involves studying geographic data, and polar coordinates can be useful for analyzing spatial patterns around a central point. For example, if you're studying the distribution of earthquakes around a volcano, you might use polar coordinates to map the locations and analyze their density as a function of distance from the volcano's center. Double integrals could be used to calculate the total area affected by a certain phenomenon within a given radius of the volcano. This type of analysis helps scientists understand natural hazards, plan urban development, and manage resources effectively.

Advanced Materials and Manufacturing

In materials science and manufacturing, polar coordinates are used to analyze the properties of materials with cylindrical symmetry, such as pipes, rods, and composite structures. For example, when analyzing the stress distribution in a cylindrical pressure vessel, engineers use double integrals in polar coordinates to calculate the forces acting on the material. This helps ensure the structural integrity and safety of the vessel. In manufacturing processes like drilling or turning, polar coordinates are used to control the motion of cutting tools and generate precise circular shapes. Consider the design of a composite material with fibers arranged in a circular pattern – polar coordinates are essential for modeling the material's mechanical properties and predicting its behavior under stress.

Conclusion: Mastering Polar Coordinates for Integral Evaluation

So, we've successfully tackled a double integral using polar coordinates! Remember, the key is to recognize when polar coordinates can simplify the problem, especially when dealing with circular regions. Practice these conversions and integrations, and you'll be a pro in no time. Plus, as we've seen, this isn't just about abstract math; it's a powerful tool with applications in diverse fields, making your mathematical skills truly valuable in the real world. Keep exploring, keep learning, and you'll be amazed at the connections you discover! Understanding these applications not only enhances your appreciation for the mathematical concepts but also prepares you for a wide range of career opportunities in STEM fields. By mastering polar coordinates and double integrals, you're equipping yourself with a versatile toolset for solving complex problems and making meaningful contributions to various industries. Keep practicing, stay curious, and the world of mathematics will continue to unfold its fascinating possibilities for you!