10.1 Double integrals over non-rectangular regions

Double integrals over non-rectangular regions expand our ability to calculate areas and volumes of complex shapes. We'll learn to identify Type I and Type II regions, sketch them, and set up iterated integrals using boundary functions.

Fubini's Theorem lets us evaluate these integrals in either order, giving us flexibility in our approach. We'll practice setting up and solving double integrals for various non-rectangular regions, building on our previous knowledge of integration.

Non-rectangular Regions

Region Types

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• Non-rectangular regions regions that cannot be described as a rectangle in the xy-plane with sides parallel to the x and y axes
• Type I region a region that can be described by $a \leq x \leq b$ and $g_1(x) \leq y \leq g_2(x)$, where $g_1$ and $g_2$ are continuous functions on $[a,b]$
• Type II region a region that can be described by $c \leq y \leq d$ and $h_1(y) \leq x \leq h_2(y)$, where $h_1$ and $h_2$ are continuous functions on $[c,d]$

Sketching Regions

• Sketch of region an important step in setting up double integrals over non-rectangular regions
  • Identify the type of region (Type I or Type II)
  • Determine the boundary functions ($g_1$, $g_2$, $h_1$, $h_2$) and their domains ($[a,b]$ or $[c,d]$)
  • Sketch the region by plotting the boundary functions and shading the area between them

Iterated Integrals Over Non-rectangular Regions

Setting Up Iterated Integrals

• Iterated integrals double integrals that are evaluated by iterating the integration process, first with respect to one variable and then with respect to the other
• Boundary functions the functions that define the limits of integration for each variable in an iterated integral
  • For Type I regions: $g_1(x)$ and $g_2(x)$
  • For Type II regions: $h_1(y)$ and $h_2(y)$
• Integration limits the values of the variables at which the integration starts and ends, determined by the boundary functions and their domains

Fubini's Theorem

• Fubini's Theorem states that if $f(x,y)$ is continuous over a closed, bounded region $R$, then the double integral of $f$ over $R$ can be evaluated using iterated integrals in either order
  • For Type I regions: $\int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y) dy dx = \int_R f(x,y) dA$
  • For Type II regions: $\int_c^d \int_{h_1(y)}^{h_2(y)} f(x,y) dx dy = \int_R f(x,y) dA$
• Fubini's Theorem allows for flexibility in choosing the order of integration based on the region type and the complexity of the integrand Examples:

1. Type I region: $R = \{(x,y) | 0 \leq x \leq 1, x^2 \leq y \leq \sqrt{x}\}$
   • Boundary functions: $g_1(x) = x^2$ and $g_2(x) = \sqrt{x}$
   • Iterated integral: $\int_0^1 \int_{x^2}^{\sqrt{x}} f(x,y) dy dx$
2. Type II region: $R = \{(x,y) | 0 \leq y \leq 1, y \leq x \leq y^2 + 1\}$
   • Boundary functions: $h_1(y) = y$ and $h_2(y) = y^2 + 1$
   • Iterated integral: $\int_0^1 \int_y^{y^2+1} f(x,y) dx dy$