Double Integrals over General Regions

This guide covers the transition from integrating over simple rectangles to integrating over general, non-rectangular regions in the $xy$-plane using iterated integrals with variable limits.

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Learning Objectives

By the end of this study guide, you should be able to:

• Recognize when a function of two variables is integrable over a general region $D$.
• Classify regions as Type I (vertically simple) or Type II (horizontally simple).
• Evaluate double integrals by setting up iterated integrals with variable boundaries.
• Simplify complex calculations by changing the order of integration.
• Apply double integrals to calculate areas of plane regions and volumes under surfaces.

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Key Terms & Glossary

• Type I Region: A region bounded on the left and right by vertical lines $x=a$ and $x=b$, and bounded above and below by continuous functions $y = g_2(x)$ and $y = g_1(x)$.
• Type II Region: A region bounded above and below by horizontal lines $y=c$ and $y=d$, and bounded on the left and right by continuous functions $x = h_1(y)$ and $x = h_2(y)$.
• Iterated Integral: An integral performed in succession (one variable at a time), where the inner limits may depend on the outer variable.
• Fubini’s Theorem (Strong Form): The theorem that allows us to evaluate double integrals over Type I or Type II regions as iterated integrals.

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The "Big Idea"

In previous sections, we integrated over rectangles where all limits were constants. However, the real world isn't made of rectangles. To integrate over a general shape $D$, we imagine it sits inside a larger rectangle $R$. We define a new function that matches our original function inside $D$ and is zero outside $D$. This allows us to use the power of calculus on any bounded shape by making the limits of integration functions themselves rather than just numbers.

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Formula / Concept Box

[!IMPORTANT] Always ensure the outer limits are constants. If your outer limits contain variables, the result will not be a scalar value, which indicates an error in setup.

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Hierarchical Outline

1. Defining the Integral over $D$
   • Extension Function: Defining $f(x,y)=0$ outside the region $D$.
   • Integrability: Smooth boundaries and continuous functions ensure the integral exists.
2. Type I Regions (Vertically Simple)
   • Description: $D = \{(x,y) \mid a \le x \le b, g_1(x) \le y \le g_2(x)\}$.
   • Integration Order: $dy$ then $dx$.
3. Type II Regions (Horizontally Simple)
   • Description: $D = \{(x,y) \mid c \le y \le d, h_1(y) \le x \le h_2(y)\}$.
   • Integration Order: $dx$ then $dy$.
4. Applications
   • Area: $\iint_D 1 \, dA$.
   • Volume: $\iint_D f(x,y) \, dA$ where $f(x,y) \ge 0$.
   • Average Value: $f_{ave} = \frac{1}{Area(D)} \iint_D f(x,y) \, dA$.

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Visual Anchors

Choosing the Order of Integration

Geometric Representation of a Type I Region

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Definition-Example Pairs

• Term: Area as a Double Integral
• Definition: The area of a region $D$ is the double integral of the constant function $f(x,y) = 1$ over that region.
• Real-World Example: Calculating the surface area of a custom-shaped swimming pool. If the pool is bounded by the curves $y=x^2$ and $y=x+2$, the area is found by $\int_{-1}^{2} \int_{x^2}^{x+2} 1 \, dy \, dx$.

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Worked Examples

Example 1: Evaluating a Type I Integral

Problem: Evaluate $\iint_D (x+2y) \, dA$ where $D$ is the region bounded by $y=x^2$ and $y=2x$.

Step-by-Step Breakdown:

1. Find Intersections: Set $x^2 = 2x \implies x(x-2)=0$. Points are $(0,0)$ and $(2,4)$.
2. Determine Type: Between $x=0$ and $x=2$, the line $y=2x$ is above the parabola $y=x^2$. This is Type I.
3. Setup Integral: $\int_{0}^{2} \int_{x^2}^{2x} (x+2y) \, dy \, dx$
4. Inner Integration ($dy$): $[xy + y^2]_{x^2}^{2x} = (x(2x) + (2x)^2) - (x(x^2) + (x^2)^2) = 6x^2 - x^3 - x^4$
5. Outer Integration ($dx$): $\int_{0}^{2} (6x^2 - x^3 - x^4) \, dx = [2x^3 - \frac{x^4}{4} - \frac{x^5}{5}]_0^2 = 16 - 4 - 6.4 = 5.6$

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Checkpoint Questions

1. What is the main difference between Fubini's theorem for rectangles and the "Strong Form" of Fubini's theorem?
2. Identify the region $D$ defined by $\int_{0}^{1} \int_{y}^{1} f(x,y) \, dx \, dy$. Is it Type I or Type II?
3. Why is it often necessary to change the order of integration for an integral like $\int_{0}^{1} \int_{x}^{1} e^{y^2} \, dy \, dx$?
4. Write the double integral expression for the average value of $f(x,y) = xy$ over a triangle with vertices $(0,0), (1,0),$ and $(0,1)$.

▶Click to reveal answers

1. The Strong Form allows the inner limits of integration to be functions of the outer variable, whereas the standard form uses only constants.
2. It is described as Type II (horizontally simple). The region is a triangle with vertices (0,0), (1,0), and (1,1).
3. Because $e^{y^2}$ has no elementary antiderivative with respect to $y$. Switching to $dx \, dy$ makes the inner integral easy.
4. $Area = 0.5$. $f_{ave} = \frac{1}{0.5} \int_{0}^{1} \int_{0}^{1-x} xy \, dy \, dx$.