 
7*y*2 *dA*,    *D* is the triangular region with vertices (0, 1), (1, 2), (4, 1)

*D*

To integrate \( 7y^2 \, dA \) over the triangular region \( D \) with vertices at \((0,1)\), \((1,2)\), and \((4,1)\), we need to establish the bounds of integration for \(y\) and \(x\) based on the geometry of \( D \).

### Step 1: Determine Equations of the Triangle's Edges
1. **Line between** \((0,1)\) **and** \((1,2)\):
   - Slope \( = \frac{2 - 1}{1 - 0} = 1\)
   - Equation: \( y = x + 1 \)

2. **Line between** \((1,2)\) **and** \((4,1)\):
   - Slope \( = \frac{1 - 2}{4 - 1} = -\frac{1}{3}\)
   - Equation: \( y = -\frac{1}{3}x + \frac{7}{3} \)

3. **Line between** \((0,1)\) **and** \((4,1)\):
   - This is a horizontal line where \( y = 1 \).

### Step 2: Set up the Bounds
For this region, \(x\) ranges from 0 to 4. For each fixed \(x\), \(y\) ranges from the lower edge (the line \(y = 1\)) to the upper edge of the triangle:
   - **Lower bound**: \( y = 1 \)
   - **Upper bound**: 
     - When \( 0 \leq x \leq 1 \), \( y = x + 1 \)
     - When \( 1 \leq x \leq 4 \), \( y = -\frac{1}{3}x + \frac{7}{3} \)

### Step 3: Set Up the Integral
The integral of \(7y^2 \, dA\) over \(D\) can be expressed as:
\[
\int_{0}^{1} \int_{1}^{x+1} 7y^2 \, dy \, dx + \int_{1}^{4} \int_{1}^{-\frac{1}{3}x + \frac{7}{3}} 7y^2 \, dy \, dx
\]

### Step 4: Solve the Integral
#### Part 1: Integral over \(0 \leq x \leq 1\)
\[
\int_{0}^{1} \int_{1}^{x+1} 7y^2 \, dy \, dx
\]
1. **Integrate with respect to \(y\)**:
   \[
   \int 7y^2 \, dy = \frac{7y^3}{3}
   \]
2. **Apply bounds \( y = 1 \) to \( y = x + 1 \)**.

#### Part 2: Integral over \(1 \leq x \leq 4\)
\[
\int_{1}^{4} \int_{1}^{-\frac{1}{3}x + \frac{7}{3}} 7y^2 \, dy \, dx
\]
1. **Integrate with respect to \(y\)**:
   \[
   \int 7y^2 \, dy = \frac{7y^3}{3}
   \]
2. **Apply bounds \( y = 1 \) to \( y = -\frac{1}{3}x + \frac{7}{3} \)**.

Combining these, compute each integral and add the results for the final answer. Would you like a detailed evaluation of these integrals?

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1. How do we choose integration bounds based on triangular regions?
2. What changes if one of the vertices shifts?
3. How do integrals handle different slopes in region boundaries?
4. How to verify the region's area matches integral results?
5. What is the effect of increasing \(y^2\) coefficient in integrand?

**Tip:** Drawing regions and boundaries can help visualize integration bounds more effectively.

Integrate 7y^2 dA, where D is the triangular region with vertices (0, 1), (1, 2), and (4, 1).

Explore the integration of 7y^2 over a triangular region defined by vertices at (0,1), (1,2), and (4,1). Learn how to determine the integration bounds for y and x based on the triangle's edges and apply multivariable calculus techniques to evaluate the integral. Suitable for college-level calculus students.

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