9.1 Definition and properties of double integrals

Double integrals extend the concept of integration to functions of two variables. They calculate the volume under a surface over a rectangular region in the xy-plane, representing a powerful tool for analyzing three-dimensional spaces and their properties.

This section introduces the definition and key properties of double integrals. We'll learn how to set up and evaluate these integrals, understand their geometric interpretation, and explore important theorems that make working with them easier and more intuitive.

Definition and Properties

Double Integral and Rectangular Region

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• Double integral represents the volume under a surface $z=f(x,y)$ over a rectangular region $R$ in the $xy$-plane
• Rectangular region $R=[a,b]\times[c,d]$ is the Cartesian product of two closed intervals $[a,b]$ and $[c,d]$
  • Can be visualized as a rectangle with sides parallel to the coordinate axes
  • Defined by the inequalities $a\leq x\leq b$ and $c\leq y\leq d$
• Double integral of a function $f(x,y)$ over a rectangular region $R$ is denoted as $\iint_R f(x,y) dA$
  • $dA$ represents the area element in the $xy$-plane
  • Geometrically, the double integral gives the volume of the solid bounded by the surface $z=f(x,y)$ and the region $R$

Integrable Function and Properties

• Function $f(x,y)$ is integrable over a rectangular region $R$ if the double integral $\iint_R f(x,y) dA$ exists
  • Integrability implies the function is bounded and has a finite number of discontinuities in $R$
• Additive property states that if $f(x,y)$ is integrable over rectangular regions $R_1$ and $R_2$ that do not overlap except possibly along their boundaries, then $f(x,y)$ is integrable over $R_1\cup R_2$ and $\iint_{R_1\cup R_2} f(x,y) dA = \iint_{R_1} f(x,y) dA + \iint_{R_2} f(x,y) dA$
  • Allows the division of a region into smaller subregions for easier computation
• Linearity property states that if $f(x,y)$ and $g(x,y)$ are integrable over a rectangular region $R$ and $c$ is a constant, then $cf(x,y)$ and $f(x,y)+g(x,y)$ are also integrable over $R$ and:
  • $\iint_R cf(x,y) dA = c\iint_R f(x,y) dA$
  • $\iint_R [f(x,y)+g(x,y)] dA = \iint_R f(x,y) dA + \iint_R g(x,y) dA$

Riemann Sums

Approximating Double Integrals

• Riemann sum approximates the double integral by partitioning the rectangular region $R$ into smaller rectangles and summing the volumes of rectangular prisms
• Partition the intervals $[a,b]$ and $[c,d]$ into $m$ and $n$ subintervals of equal width $\Delta x = \frac{b-a}{m}$ and $\Delta y = \frac{d-c}{n}$, respectively
  • This creates $mn$ subrectangles $R_{ij}$ with dimensions $\Delta x$ by $\Delta y$
• Choose a sample point $(x_{ij}^*,y_{ij}^*)$ within each subrectangle $R_{ij}$
• Riemann sum is given by $\sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*,y_{ij}^*) \Delta x \Delta y$
  • Represents the sum of the volumes of rectangular prisms with base area $\Delta x \Delta y$ and height $f(x_{ij}^*,y_{ij}^*)$

Limit of Riemann Sum

• As the number of partitions $m$ and $n$ approach infinity, the Riemann sum approaches the exact value of the double integral
• Limit of the Riemann sum is written as $\lim_{m,n\to\infty} \sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*,y_{ij}^*) \Delta x \Delta y = \iint_R f(x,y) dA$
  • This limit exists if and only if the function $f(x,y)$ is integrable over the rectangular region $R$
• The choice of sample points $(x_{ij}^*,y_{ij}^*)$ does not affect the limit as long as they lie within their respective subrectangles $R_{ij}$
  • Common choices include the midpoint, lower left corner, or upper right corner of each subrectangle

Theorems

Comparison Theorem for Double Integrals

• Comparison theorem states that if $f(x,y) \leq g(x,y)$ for all $(x,y)$ in a rectangular region $R$, then $\iint_R f(x,y) dA \leq \iint_R g(x,y) dA$
  • Provides a way to estimate or bound the value of a double integral
• If $f(x,y) \leq g(x,y) \leq h(x,y)$ for all $(x,y)$ in $R$, then $\iint_R f(x,y) dA \leq \iint_R g(x,y) dA \leq \iint_R h(x,y) dA$
  • Sandwich theorem for double integrals
• Useful when the exact value of a double integral is difficult to compute but can be bounded by simpler functions
  • For example, if $0 \leq f(x,y) \leq 1$ over a rectangular region $R$ with area $A$, then $0 \leq \iint_R f(x,y) dA \leq A$