10.3 Applications to area and volume

Double integrals over general regions are powerful tools for calculating areas and volumes. They let us handle complex shapes by breaking them down into smaller pieces and adding them up.

This section focuses on applying double integrals to find areas of 2D regions and volumes of 3D solids. We'll learn how to set up integrals, choose coordinate systems, and interpret results for real-world problems.

Area and Volume Calculation

Calculating Area Using Double Integrals

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• Calculate the area of a region in the xy-plane by setting up a double integral over the region
  • The double integral $\iint_{R} dA$ gives the area of the region $R$
  • The limits of integration depend on the shape of the region (rectangular, non-rectangular, or polar coordinates)
• Convert between rectangular and polar coordinates when calculating areas
  • Rectangular coordinates: $dA = dxdy$
  • Polar coordinates: $dA = rdrd\theta$
• Determine the order of integration based on the region's boundaries
  • Integrate with respect to $y$ first if the region is bounded by functions of $x$
  • Integrate with respect to $x$ first if the region is bounded by functions of $y$

Volume Calculation Using Double Integrals

• Calculate the volume of a solid region by integrating a function $f(x,y)$ over a region $R$ in the xy-plane
  • The double integral $\iint_{R} f(x,y) dA$ gives the volume of the solid region
  • The function $f(x,y)$ represents the height of the solid at each point $(x,y)$ in the region $R$
• Set up the limits of integration based on the region's boundaries in the xy-plane
• Determine the order of integration based on the region's boundaries
  • Integrate with respect to $y$ first if the region is bounded by functions of $x$
  • Integrate with respect to $x$ first if the region is bounded by functions of $y$

Surface Area and Solid Regions

• Calculate the surface area of a solid by integrating over the region in the xy-plane
  • The surface area is given by the double integral $\iint_{R} \sqrt{1 + (\frac{\partial z}{\partial x})^2 + (\frac{\partial z}{\partial y})^2} dA$
  • The function $z = f(x,y)$ represents the surface of the solid
• Determine the boundaries of the solid region in the xy-plane
  • The solid region is the projection of the surface onto the xy-plane
• Set up the limits of integration based on the boundaries of the solid region
• Calculate the volume of a solid region using the method of disks or shells
  • Method of disks: Integrate the cross-sectional area of the solid (disks) along the axis of rotation
  • Method of shells: Integrate the lateral surface area of the solid (shells) along the axis of rotation

Physical Properties

Moment of Inertia

• Calculate the moment of inertia of a planar region using double integrals
  • The moment of inertia measures the resistance of an object to rotational acceleration
  • For a planar region with density $\rho(x,y)$, the moment of inertia about the x-axis is $I_x = \iint_{R} y^2 \rho(x,y) dA$
  • For a planar region with density $\rho(x,y)$, the moment of inertia about the y-axis is $I_y = \iint_{R} x^2 \rho(x,y) dA$
• Determine the region $R$ over which the double integral is evaluated
• Set up the limits of integration based on the region's boundaries

Center of Mass

• Find the center of mass (centroid) of a planar region using double integrals
  • The center of mass is the point where the object's mass is evenly distributed
  • For a planar region with density $\rho(x,y)$, the x-coordinate of the center of mass is $\bar{x} = \frac{\iint_{R} x \rho(x,y) dA}{\iint_{R} \rho(x,y) dA}$
  • For a planar region with density $\rho(x,y)$, the y-coordinate of the center of mass is $\bar{y} = \frac{\iint_{R} y \rho(x,y) dA}{\iint_{R} \rho(x,y) dA}$
• Determine the region $R$ over which the double integrals are evaluated
• Set up the limits of integration based on the region's boundaries

Density Functions

• Understand the concept of density functions in the context of double integrals
  • A density function $\rho(x,y)$ describes the mass per unit area at each point $(x,y)$ in a region
  • The total mass of a planar region with density $\rho(x,y)$ is given by the double integral $\iint_{R} \rho(x,y) dA$
• Incorporate density functions into calculations of physical properties
  • Density functions are used in the calculation of moments of inertia and centers of mass
• Determine the appropriate density function based on the given problem
  • Constant density: $\rho(x,y) = \rho_0$
  • Variable density: $\rho(x,y)$ is a function of $x$ and $y$

Advanced Geometry

Curved Surfaces

• Parameterize curved surfaces using vector-valued functions
  • A curved surface can be represented by a vector-valued function $\vec{r}(u,v) = \langle x(u,v), y(u,v), z(u,v) \rangle$
  • The parameters $u$ and $v$ vary over a region $D$ in the uv-plane
• Calculate surface integrals over curved surfaces
  • A surface integral of a function $f(x,y,z)$ over a curved surface $S$ is given by $\iint_{S} f(x,y,z) dS$
  • The surface element $dS$ is determined by the cross product of the partial derivatives of the vector-valued function: $dS = \left\| \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v} \right\| dudv$
• Apply surface integrals to find the area of a curved surface
  • The area of a curved surface $S$ is given by the surface integral $\iint_{S} dS$
• Determine the limits of integration based on the region $D$ in the parameter space
  • The region $D$ in the uv-plane corresponds to the domain of the vector-valued function $\vec{r}(u,v)$