5.2 Double Integrals over General Regions

Double integrals extend the concept of integration to two dimensions. They allow us to calculate quantities like volume, mass, and average value over regions in the plane. Understanding how to set up and evaluate these integrals is crucial for many applications in physics and engineering.

Techniques for evaluating double integrals include changing the order of integration, using polar coordinates, and leveraging symmetry. These methods can simplify complex integrals and make them more manageable to solve. Mastering these techniques opens up a wide range of problem-solving possibilities in multivariable calculus.

Double Integrals over General Regions

Integrable functions over regions

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• A function $f(x, y)$ is integrable over a region $R$ if it is continuous or has a finite number of discontinuities within the region (removable, jump, or infinite discontinuities)
• The region $R$ must be bounded meaning it can be enclosed within a rectangular region with boundaries defined by functions or curves

Iterated integrals for bounded regions

• To evaluate a double integral $\iint_R f(x, y) [dA](https://www.fiveableKeyTerm:dA)$, use iterated integrals
  • Determine the order of integration ($dy$ then $dx$, or $dx$ then $dy$)
  • Set up the outer integral with respect to one variable ($x$) and the inner integral with respect to the other variable ($y$)
  • The inner integral limits are functions of the outer variable
  • The outer integral limits are constants
• $\int_a^b \int_{g_1(x)}^{g_2(x)} f(x, y) dy dx$, where $g_1(x)$ and $g_2(x)$ are the lower and upper bounds of $y$ for a given $x$
• The region of integration can be classified as a type I region (bounded by vertical lines) or a type II region (bounded by horizontal lines)

Order of integration optimization

• Changing the order of integration can simplify the integral or make it easier to evaluate
  • Sketch the region $R$ and identify the bounds for each variable
  • Determine the new order of integration and set up the iterated integral accordingly
  • Adjust the integration limits to reflect the new order
• If $\int_a^b \int_{g_1(x)}^{g_2(x)} f(x, y) dy dx$ is difficult to evaluate, try $\int_c^d \int_{h_1(y)}^{h_2(y)} f(x, y) dx dy$, where $h_1(y)$ and $h_2(y)$ are the lower and upper bounds of $x$ for a given $y$

Applications of double integrals

• Volume under a surface $z = f(x, y)$ over a region $R$: $V = \iint_R f(x, y) dA$
• Area of a region $R$ in the $xy$-plane: $A = \iint_R dA$
• Average value of a function $f(x, y)$ over a region $R$: $\frac{1}{A} \iint_R f(x, y) dA$, where $A$ is the area of the region $R$

Double improper integrals

• A double improper integral has at least one infinite limit of integration or an integrand that is unbounded within the region of integration
• To evaluate a double improper integral:
  1. Set up the iterated integral with finite limits of integration
  2. Evaluate the inner integral
  3. Take the limit of the outer integral as the appropriate limit(s) approach infinity or the point(s) of unboundedness
• If the limit exists and is finite, the improper integral converges; otherwise, it diverges
• $\int_0^{\infty} \int_0^{\infty} \frac{e^{-(x+y)}}{x+y} dy dx$

Techniques for Evaluating Double Integrals

Convert between rectangular and polar coordinates

• Converting from rectangular coordinates $(x, y)$ to polar coordinates $(r, \theta)$:
  • $x = r \cos \theta$
  • $y = r \sin \theta$
  • $dA = r dr d\theta$
• Converting from polar coordinates $(r, \theta)$ to rectangular coordinates $(x, y)$:
  • $r = \sqrt{x^2 + y^2}$
  • $\theta = \arctan(\frac{y}{x})$, with adjustments based on the quadrant
• Double integrals in polar coordinates: $\iint_R f(x, y) dA = \iint_R f(r \cos \theta, r \sin \theta) r dr d\theta$

Use symmetry to simplify double integrals

• If a region $R$ and the integrand $f(x, y)$ are symmetric about the $x$-axis, $y$-axis, or origin, the double integral can be simplified
  • Symmetry about the $x$-axis: $\iint_R f(x, y) dA = 2 \iint_{R_1} f(x, y) dA$, where $R_1$ is the portion of $R$ above the $x$-axis
  • Symmetry about the $y$-axis: $\iint_R f(x, y) dA = 2 \iint_{R_2} f(x, y) dA$, where $R_2$ is the portion of $R$ to the right of the $y$-axis
  • Symmetry about the origin: $\iint_R f(x, y) dA = 4 \iint_{R_3} f(x, y) dA$, where $R_3$ is the portion of $R$ in the first quadrant

Advanced Techniques and Concepts

• Green's theorem relates a line integral around a simple closed curve to a double integral over the region enclosed by the curve
• Parametric equations can be used to describe curves that form the boundaries of integration regions
• Level curves of a function $f(x, y)$ are the set of points where $f(x, y) = k$ for some constant $k$, useful for visualizing functions of two variables