Recall what we discussed in class by regarding a certain domain as either type 1 or type 2 domain. Evaluate (r+y) dA, where the region D is enclosed by the triangle with vertices (0,0), (1, 1), (2,0).
Recall what we discussed in class by regarding a certain domain as either type 1 or type 2 domain. Evaluate (r+y) dA, where the region D is enclosed by the triangle with vertices (0,0), (1, 1), (2,0).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Instruction for Evaluating Double Integral over a Triangular Region**
Recall what we discussed in class regarding a certain domain as either a type 1 or type 2 domain. Your task is to evaluate the double integral:
\[
\iint_D (x + y) \, dA
\]
where the region \( D \) is enclosed by the triangle with vertices \((0, 0)\), \((1, 1)\), and \((2, 0)\).
**Steps to Solution:**
1. **Determining the Region \(D\):**
- Plot the vertices on a coordinate plane to form a triangle.
- The base of the triangle is along the x-axis from \((0, 0)\) to \((2, 0)\).
- The diagonal spans from \((0, 0)\) to \((1, 1)\).
- The triangle is bounded by:
- Line 1: \( y = x \) (from \((0, 0)\) to \((1, 1)\))
- Line 2: \( y = -x + 2 \) (from \((2, 0)\) to \((1, 1)\))
2. **Evaluating the Integral:**
- Depending on choosing type 1 or type 2 domain, set up the limits of integration accordingly.
- For a type 1 region \( y \) depends on \( x \):
- Integrate with respect to \( y \) first, from \( y = x \) to \( y = -x + 2 \).
- Then integrate \( x \) from 0 to 2.
- For a type 2 region \( x \) depends on \( y \):
- Integrate with respect to \( x \) first, from \( x = y \) to \( x = 2 - y \).
- Then integrate \( y \) from 0 to 1.
This exercise will help consolidate your understanding of setting up and evaluating double integrals over triangular areas using different types of domains.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1d98f491-3934-4249-9e36-e6d1c5a7d193%2F16952e2d-1a84-476e-a3e0-2fb4e6ecbc27%2Fecn42yf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Instruction for Evaluating Double Integral over a Triangular Region**
Recall what we discussed in class regarding a certain domain as either a type 1 or type 2 domain. Your task is to evaluate the double integral:
\[
\iint_D (x + y) \, dA
\]
where the region \( D \) is enclosed by the triangle with vertices \((0, 0)\), \((1, 1)\), and \((2, 0)\).
**Steps to Solution:**
1. **Determining the Region \(D\):**
- Plot the vertices on a coordinate plane to form a triangle.
- The base of the triangle is along the x-axis from \((0, 0)\) to \((2, 0)\).
- The diagonal spans from \((0, 0)\) to \((1, 1)\).
- The triangle is bounded by:
- Line 1: \( y = x \) (from \((0, 0)\) to \((1, 1)\))
- Line 2: \( y = -x + 2 \) (from \((2, 0)\) to \((1, 1)\))
2. **Evaluating the Integral:**
- Depending on choosing type 1 or type 2 domain, set up the limits of integration accordingly.
- For a type 1 region \( y \) depends on \( x \):
- Integrate with respect to \( y \) first, from \( y = x \) to \( y = -x + 2 \).
- Then integrate \( x \) from 0 to 2.
- For a type 2 region \( x \) depends on \( y \):
- Integrate with respect to \( x \) first, from \( x = y \) to \( x = 2 - y \).
- Then integrate \( y \) from 0 to 1.
This exercise will help consolidate your understanding of setting up and evaluating double integrals over triangular areas using different types of domains.
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