∞Calculus IV Unit 11 – Double Integrals in Polar Coordinates

Key Concepts

• Double integrals in polar coordinates evaluate a function over a two-dimensional region using polar coordinates $(r, \theta)$
• Polar coordinates consist of a radial distance $r$ and an angle $\theta$ measured counterclockwise from the positive $x$-axis
• Converting between Cartesian $(x, y)$ and polar $(r, \theta)$ coordinates uses the relationships $x = r\cos\theta$ and $y = r\sin\theta$
• The area element in polar coordinates is $dA = r \, dr \, d\theta$, which replaces $dA = dx \, dy$ in Cartesian coordinates
• The limits of integration in polar coordinates depend on the shape of the region and are often functions of $\theta$
• Polar double integrals are particularly useful for regions with circular symmetry or those more easily described in polar coordinates
• The order of integration in polar coordinates is typically $\int_a^b \int_{g_1(\theta)}^{g_2(\theta)} f(r, \theta) \, r \, dr \, d\theta$, where $a$ and $b$ are the limits for $\theta$, and $g_1(\theta)$ and $g_2(\theta)$ are the limits for $r$

Polar Coordinate Basics

• Polar coordinates $(r, \theta)$ describe a point's position using a distance $r$ from the origin and an angle $\theta$ from the positive $x$-axis
  • $r$ is the radial coordinate and can take on values from 0 to $\infty$
  • $\theta$ is the angular coordinate and is typically measured in radians, with a range of $0 \leq \theta < 2\pi$
• The polar coordinate system is centered at the origin $(0, 0)$, which corresponds to $r = 0$
• Points with the same angle $\theta$ lie on a straight line emanating from the origin, called a radial line
• Points with the same distance $r$ from the origin lie on a circle centered at the origin with radius $r$
• Polar coordinates are useful for describing curves and regions with circular symmetry, such as circles, spirals, and rosettes
• The polar coordinate system is periodic in $\theta$, meaning that $(r, \theta)$ and $(r, \theta + 2\pi)$ represent the same point

Converting Between Cartesian and Polar

• To convert from Cartesian coordinates $(x, y)$ to polar coordinates $(r, \theta)$, use the equations:
  • $r = \sqrt{x^2 + y^2}$
  • $\theta = \tan^{-1}(\frac{y}{x})$, with adjustments based on the quadrant
• To convert from polar coordinates $(r, \theta)$ to Cartesian coordinates $(x, y)$, use the equations:
  • $x = r\cos\theta$
  • $y = r\sin\theta$
• When converting from polar to Cartesian, be mindful of the quadrant based on the signs of $x$ and $y$
  • Quadrant I: $x > 0$, $y > 0$
  • Quadrant II: $x < 0$, $y > 0$
  • Quadrant III: $x < 0$, $y < 0$
  • Quadrant IV: $x > 0$, $y < 0$
• Some common polar equations and their Cartesian counterparts include:
  • Circle: $r = a$ (polar) $\leftrightarrow$ $x^2 + y^2 = a^2$ (Cartesian)
  • Line: $\theta = a$ (polar) $\leftrightarrow$ $y = x\tan a$ (Cartesian)
• Practice converting between the two coordinate systems to build familiarity and intuition

Setting Up Double Integrals in Polar Form

• To set up a double integral in polar coordinates, follow these steps:
  1. Sketch the region of integration in the $xy$-plane
  2. Identify the boundaries of the region in terms of polar coordinates $(r, \theta)$
  3. Determine the limits of integration for $\theta$ based on the angular extent of the region
  4. Express the limits of integration for $r$ as functions of $\theta$, denoted as $g_1(\theta)$ and $g_2(\theta)$
  5. Replace $dA = dx \, dy$ with $dA = r \, dr \, d\theta$ in the integral
• The general form of a double integral in polar coordinates is: $\iint_D f(r, \theta) \, r \, dr \, d\theta = \int_a^b \int_{g_1(\theta)}^{g_2(\theta)} f(r, \theta) \, r \, dr \, d\theta$
• When setting up the limits of integration, consider the following:
  • For a full circle with radius $a$, the limits are typically $0 \leq \theta \leq 2\pi$ and $0 \leq r \leq a$
  • For a sector of a circle, the limits for $\theta$ will be a subinterval of $[0, 2\pi]$, such as $0 \leq \theta \leq \frac{\pi}{4}$
  • For more complex regions, the limits for $r$ may be functions of $\theta$, such as $a \leq r \leq b\cos\theta$
• When the region of integration is symmetric about the origin or an axis, consider simplifying the integral by exploiting the symmetry
• Practice setting up double integrals in polar coordinates for various regions to develop proficiency

Evaluating Polar Double Integrals

• To evaluate a double integral in polar coordinates, follow these steps:
  1. Set up the integral with the appropriate limits of integration and the polar area element $dA = r \, dr \, d\theta$
  2. Integrate with respect to $r$ first, treating $\theta$ as a constant
  3. Integrate the resulting expression with respect to $\theta$
  4. Evaluate the integral and simplify the final result
• When integrating with respect to $r$, use common integration techniques such as substitution, integration by parts, or trigonometric identities as needed
• When integrating with respect to $\theta$, be mindful of the periodicity of trigonometric functions and use trigonometric identities to simplify expressions
• If the region of integration is symmetric about the origin or an axis, consider simplifying the integral before evaluating
  • For example, if the region is symmetric about the $x$-axis, you can evaluate the integral over the upper half of the region and multiply the result by 2
• Pay attention to the units of the final result, which may involve square units for area or cubic units for volume
• Practice evaluating polar double integrals for various functions and regions to build confidence and identify common patterns

Applications and Examples

• Double integrals in polar coordinates have numerous applications in mathematics, physics, and engineering, such as:
  • Calculating areas and volumes of regions with circular symmetry
  • Computing moments of inertia for objects with radial density functions
  • Evaluating probability density functions in polar coordinates
  • Solving problems involving electric and magnetic fields with radial symmetry
• Example: Find the area of the region enclosed by the circle $r = 2\cos\theta$
  • Set up the integral: $A = \int_0^{2\pi} \int_0^{2\cos\theta} r \, dr \, d\theta$
  • Evaluate the integral: $A = \int_0^{2\pi} \cos^2\theta \, d\theta = \pi$
• Example: Calculate the volume of a solid obtained by rotating the region bounded by $r = \sin\theta$ and $r = \cos\theta$ about the $x$-axis
  • Set up the integral: $V = \int_0^{\frac{\pi}{4}} \int_{\sin\theta}^{\cos\theta} 2\pi r^2 \sin\theta \, dr \, d\theta$
  • Evaluate the integral: $V = \frac{\pi}{12}(2\sqrt{2} - 1)$
• Example: Find the moment of inertia of a disk with radius $a$ and radially varying density $\rho(r) = kr^2$, where $k$ is a constant
  • Set up the integral: $I = \int_0^{2\pi} \int_0^a kr^4 \, dr \, d\theta$
  • Evaluate the integral: $I = \frac{1}{2}ka^5\pi$
• Analyze and discuss the results of these examples to deepen your understanding of the applications of polar double integrals

Common Challenges and Tips

• Sketching the region of integration in polar coordinates can be challenging, especially for more complex boundaries
  • Practice sketching various regions and curves in polar coordinates to develop a strong visual intuition
  • Pay attention to the symmetry of the region and how it relates to the limits of integration
• Determining the limits of integration, particularly when they are functions of $\theta$, can be tricky
  • Carefully analyze the boundaries of the region and express them in terms of polar coordinates
  • Consider breaking the region into smaller subregions with simpler boundaries and integrating over each subregion separately
• Integrating with respect to $\theta$ may involve trigonometric functions and identities
  • Brush up on your trigonometric integration techniques and common identities
  • Look for opportunities to simplify expressions using trigonometric identities before integrating
• Keep in mind the periodicity of trigonometric functions when determining the limits of integration for $\theta$
  • Remember that $\sin(\theta + 2\pi) = \sin\theta$ and $\cos(\theta + 2\pi) = \cos\theta$
  • Adjust the limits of integration accordingly to avoid redundant calculations
• Double-check your setup and calculations, especially the limits of integration and the polar area element
  • Ensure that the limits of integration are consistent with the region being integrated over
  • Verify that the polar area element $dA = r \, dr \, d\theta$ is used correctly in the integral
• Practice, practice, practice! Work through a variety of problems to build confidence and expose yourself to different scenarios
  • Start with simpler regions and functions and gradually progress to more complex ones
  • Analyze your mistakes and learn from them to avoid repeating the same errors in the future

Practice Problems

1. Evaluate the double integral $\iint_D (r\cos\theta + r\sin\theta) \, dA$, where $D$ is the region bounded by $r = 2\sin\theta$ and $r = 2\cos\theta$ in the first quadrant.

2. Find the volume of the solid generated by rotating the region enclosed by the curves $r = 2$ and $r = 2\sin\theta$ about the $y$-axis.

3. Calculate the area of the region bounded by the curves $r = 1 + \cos\theta$ and $r = 2$.

4. Evaluate the double integral $\iint_D \frac{1}{r} \, dA$, where $D$ is the region in the first quadrant bounded by $r = 2\cos\theta$, $r = 2\sin\theta$, and $\theta = \frac{\pi}{4}$.

5. Find the moment of inertia about the $z$-axis for a thin plate in the shape of the region bounded by $r = 2\cos(2\theta)$ with density $\rho(r, \theta) = r\sin\theta$.

6. Evaluate the double integral $\iint_D r^2\cos\theta \, dA$, where $D$ is the region bounded by $r = \sin(2\theta)$ and $r = \sin\theta$ for $0 \leq \theta \leq \frac{\pi}{2}$.

7. Calculate the area of the region enclosed by the curves $r = 1 + \sin\theta$ and $r = 1 - \sin\theta$.

8. Find the volume of the solid generated by rotating the region bounded by $r = \theta$ and $r = \sin\theta$ for $0 \leq \theta \leq \frac{\pi}{2}$ about the $x$-axis.

Remember to set up the integrals properly, determine the correct limits of integration, and evaluate the integrals using the techniques discussed in the previous sections. Compare your results with the solutions and analyze any discrepancies to reinforce your understanding of double integrals in polar coordinates.