Double Integrals over General Regions
What happens when the base isn't a rectangle?
Introduction
Most real-world domains aren't rectangles. We need to integrate over disks, triangles, and blobs.
The key is to describe the boundaries as functions. Variable limits of integration will appear on the inner integral.
Type I and Type II
Type I (Vertical Simple)
We shoot vertical arrows through the region.
- x is bounded by constants: .
- y is bounded by functions: .
- Order: .
Type II (Horizontal Simple)
We shoot horizontal arrows through the region.
- y is bounded by constants: .
- x is bounded by functions: .
- Order: .
Visualizing a Type I Region
Interactive: Region between x and x^2
Worked Examples
Example 1: Type I (Triangle)
Evaluate where R is the triangle with vertices .
1. Define Boundaries:
Vertical arrows enter at and exit at .
x ranges from 0 to 1.
Setup: .
2. Integrate Inner:
.
3. Integrate Outer:
.
Example 2: Type II (Parabola)
Evaluate where D is bounded by and in the first quadrant.
Choice: Type II (Horizontal)
is the right boundary. The left is .
y goes from 0 to 4.
Setup: .
Execute:
Inner: .
Outer: .
.
Example 3: Switching Order
Evaluate .
1. Draw the Region:
Type I description: .
This is a triangle bounded by , , and .
2. Switch to Type II:
Horizontal strips: Left boundary , Right boundary .
y goes from 0 to 1.
New Integral: .
3. Solve:
Inner: .
Outer: .
Sub .
.
Practice Quiz
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