∞Calculus IV Unit 12 – Applications of Double Integrals

Key Concepts and Definitions

• Double integrals extend the concept of single integrals to functions of two variables
• Iterated integrals evaluate double integrals by integrating with respect to one variable at a time
  • Compute the inner integral first, treating the other variable as a constant
  • Then evaluate the outer integral using the result from the inner integral
• Fubini's Theorem states that if $f(x,y)$ is continuous over a closed, bounded region $R$, the double integral of $f$ over $R$ equals the iterated integral
• The order of integration can be interchanged if the function is continuous and the region is bounded
• Jacobian determinant is used when changing variables in double integrals
  • For polar coordinates, the Jacobian is $r$, so $dA = r \, dr \, d\theta$
• Double integrals can be used to find volumes, areas, centers of mass, and moments of inertia

Double Integrals in Rectangular Coordinates

• Double integrals in rectangular coordinates are written as $\iint_{R} f(x,y) \, dA$
• The region $R$ is typically described by the bounds of $x$ and $y$
• Evaluate the double integral by iterating the integrals with respect to $x$ and $y$
  • Choose the order of integration based on the region's description
  • Sketch the region to determine the appropriate bounds for each integral
• Example: $\int_{0}^{1} \int_{0}^{1-x} xy \, dy \, dx$ integrates $xy$ over the triangular region bounded by $y=0$, $x=0$, and $x+y=1$
• To change the order of integration, adjust the bounds accordingly
  • For the example above, changing to $dy \, dx$ yields $\int_{0}^{1} \int_{0}^{1-y} xy \, dx \, dy$

Double Integrals in Polar Coordinates

• Double integrals in polar coordinates are written as $\iint_{R} f(r,\theta) \, r \, dr \, d\theta$
• The region $R$ is typically described by the bounds of $r$ and $\theta$
• Convert the function $f(x,y)$ to polar form $f(r,\theta)$ using $x = r\cos\theta$ and $y = r\sin\theta$
• Determine the bounds for $r$ and $\theta$ based on the region's description in polar coordinates
  • Sketch the region to identify the appropriate bounds
• Evaluate the double integral by iterating the integrals with respect to $r$ and $\theta$
• Example: $\int_{0}^{\pi/2} \int_{0}^{1} r^2 \, dr \, d\theta$ integrates $r^2$ over the quarter circle of radius 1 in the first quadrant
• Polar coordinates simplify double integrals for regions with circular symmetry

Applications in Area Calculation

• Double integrals can be used to calculate areas of regions in the plane
• For a region $R$, the area is given by $\iint_{R} 1 \, dA$
• In rectangular coordinates, $\iint_{R} 1 \, dA = \int_{a}^{b} \int_{g_1(x)}^{g_2(x)} 1 \, dy \, dx$
  • $g_1(x)$ and $g_2(x)$ are the lower and upper bounds of $y$ in terms of $x$
• In polar coordinates, $\iint_{R} 1 \, dA = \int_{\alpha}^{\beta} \int_{h_1(\theta)}^{h_2(\theta)} r \, dr \, d\theta$
  • $h_1(\theta)$ and $h_2(\theta)$ are the lower and upper bounds of $r$ in terms of $\theta$
• Example: Find the area enclosed by the circle $x^2 + y^2 = 4$
  • In polar coordinates, the equation becomes $r^2 = 4$, so $r = 2$
  • $\int_{0}^{2\pi} \int_{0}^{2} r \, dr \, d\theta = 4\pi$

Volume Calculation Using Double Integrals

• Double integrals can calculate volumes of solids by integrating cross-sectional areas
• For a solid bounded by $z = f(x,y)$ and the region $R$ in the $xy$-plane, the volume is $\iint_{R} f(x,y) \, dA$
• In rectangular coordinates, $\iint_{R} f(x,y) \, dA = \int_{a}^{b} \int_{c}^{d} f(x,y) \, dy \, dx$
  • Integrate the cross-sectional area $f(x,y)$ over the region $R$
• In polar coordinates, $\iint_{R} f(r,\theta) \, dA = \int_{\alpha}^{\beta} \int_{g_1(\theta)}^{g_2(\theta)} f(r,\theta) \, r \, dr \, d\theta$
  • Convert $f(x,y)$ to polar form $f(r,\theta)$ and integrate over the region $R$
• Example: Find the volume of the solid bounded by $z = x^2 + y^2$ and $z = 4$
  • In rectangular coordinates, $\int_{-2}^{2} \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} (4 - x^2 - y^2) \, dy \, dx$

Center of Mass and Moments of Inertia

• Double integrals can find the center of mass and moments of inertia for thin plates
• For a thin plate with density $\delta(x,y)$ over a region $R$, the center of mass is $(\bar{x}, \bar{y})$:
  • $\bar{x} = \frac{\iint_{R} x \, \delta(x,y) \, dA}{\iint_{R} \delta(x,y) \, dA}$ and $\bar{y} = \frac{\iint_{R} y \, \delta(x,y) \, dA}{\iint_{R} \delta(x,y) \, dA}$
• Moments of inertia measure a plate's resistance to rotational acceleration
  • $I_x = \iint_{R} y^2 \, \delta(x,y) \, dA$ and $I_y = \iint_{R} x^2 \, \delta(x,y) \, dA$
• For a plate with constant density $\delta$, the formulas simplify to:
  • $\bar{x} = \frac{\iint_{R} x \, dA}{\iint_{R} dA}$, $\bar{y} = \frac{\iint_{R} y \, dA}{\iint_{R} dA}$, $I_x = \delta \iint_{R} y^2 \, dA$, $I_y = \delta \iint_{R} x^2 \, dA$

Real-World Applications and Examples

• Double integrals have numerous applications in physics, engineering, and other fields
• Calculating the mass of a thin plate with variable density $\delta(x,y)$
  • Mass $= \iint_{R} \delta(x,y) \, dA$
• Finding the average value of a function $f(x,y)$ over a region $R$
  • Average value $= \frac{\iint_{R} f(x,y) \, dA}{\iint_{R} dA}$
• Determining the electric charge on a plate with charge density $\rho(x,y)$
  • Total charge $= \iint_{R} \rho(x,y) \, dA$
• Calculating the gravitational or electrostatic potential at a point due to a thin plate
  • Potential $\propto \iint_{R} \frac{\delta(x,y)}{r} \, dA$, where $r$ is the distance from the point to $(x,y)$
• Example: Find the mass of a circular plate of radius 2 with density $\delta(x,y) = x^2 + y^2$
  • In polar coordinates, $\int_{0}^{2\pi} \int_{0}^{2} r^2 \cdot r \, dr \, d\theta = \frac{16\pi}{3}$

Common Challenges and Problem-Solving Strategies

• Setting up the correct bounds of integration based on the region's description
  • Sketch the region to visualize the bounds
  • Write the bounds in terms of the integration variables
• Choosing the appropriate coordinate system (rectangular or polar) for the problem
  • Polar coordinates are often easier for regions with circular symmetry
  • Rectangular coordinates are better suited for regions bounded by straight lines
• Evaluating integrals involving trigonometric or exponential functions
  • Use trigonometric identities and integration techniques (substitution, integration by parts)
• Remembering to include the Jacobian determinant when changing variables
  • In polar coordinates, multiply the integrand by $r$ to account for the Jacobian
• Verifying that the integrand and region satisfy the conditions of Fubini's Theorem
  • Ensure the function is continuous over the closed, bounded region
• Applying double integrals to real-world problems
  • Identify the physical quantity to be calculated (area, volume, mass, etc.)
  • Set up the integral with the appropriate integrand and region