2.2 Multiple Integrals and Applications

Multiple integrals let us calculate quantities over complex shapes in two or three dimensions. They're super useful for finding volumes, masses, and centroids of objects with varying densities or irregular shapes.

We'll learn how to set up and solve these integrals using different techniques. We'll cover iterative integration, coordinate transformations, and Jacobian determinants to simplify tricky problems and make calculations easier.

Multiple Integrals

Double and triple integrals

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• Double integrals evaluate a function over a two-dimensional region (rectangular or non-rectangular)
  • Rectangular regions integrate with respect to x first, then y, with limits of integration as constants or functions of one variable (e.g., $\int_a^b \int_c^d f(x,y) dy dx$)
  • Non-rectangular regions require determining the limits of integration based on the region's boundaries, possibly splitting the region into multiple parts (e.g., a circular region)
• Triple integrals evaluate a function over a three-dimensional region (rectangular or non-rectangular)
  • Rectangular regions integrate with respect to x, then y, and finally z, with limits of integration as constants or functions of one or two variables (e.g., $\int_a^b \int_c^d \int_e^f f(x,y,z) dz dy dx$)
  • Non-rectangular regions require determining the limits of integration based on the region's boundaries, possibly splitting the region into multiple parts
  • Cylindrical or spherical coordinates can simplify the integration process for certain regions (e.g., a spherical region)

Applications of multiple integrals

• Volume calculation uses triple integrals to find the volume of a three-dimensional object ($V = \iiint_D dV = \iiint_D dxdydz$, where $D$ is the region of integration)
• Mass calculation uses triple integrals with a density function $\rho(x, y, z)$ to find the mass of a three-dimensional object ($M = \iiint_D \rho(x, y, z) dV$)
• Centroid calculation uses triple integrals with a density function $\rho(x, y, z)$ to find the centroid coordinates $(\bar{x}, \bar{y}, \bar{z})$ of a three-dimensional object
  • $\bar{x} = \frac{1}{M} \iiint_D x\rho(x, y, z) dV$
  • $\bar{y} = \frac{1}{M} \iiint_D y\rho(x, y, z) dV$
  • $\bar{z} = \frac{1}{M} \iiint_D z\rho(x, y, z) dV$

Techniques for Solving Multiple Integrals

Iterative integration for multiple integrals

• Fubini's Theorem allows for iterative integration, where the order of integration can be changed if the integrand is continuous over the region
• Iterated integrals evaluate the innermost integral first, treating outer variables as constants, and repeat the process for the remaining integrals
• Simplifying the integrand when possible (factoring, trigonometric identities, algebraic manipulation) can make the integration process easier

Jacobian determinants in integration

• Jacobian determinant relates the area or volume elements between different coordinate systems
• For a transformation $(u, v) = (u(x, y), v(x, y))$:
  1. $dxdy = |J(u, v)| dudv$
  2. $J(u, v) = \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix}$
• For a transformation $(u, v, w) = (u(x, y, z), v(x, y, z), w(x, y, z))$:
  1. $dxdydz = |J(u, v, w)| dudvdw$
  2. $J(u, v, w) = \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} \\ \frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} & \frac{\partial w}{\partial z} \end{vmatrix}$
• Change of variables simplifies the region of integration or the integrand using common transformations (polar, cylindrical, and spherical coordinates)