https://file.aiquickdraw.com/mathbot/file/1729291816441pze0k7jm.jpg

To solve this problem, we need to compute the double integral of the function \( f(x, y) = 4 - 8x \) over the triangular region defined by the vertices \( O = (0, 0) \), \( A = (3, 7) \), and \( B = (5, 7) \).

### Step-by-Step Solution:

1. **Identifying the region of integration**:
   The vertices define a triangle, and we need to describe the bounds for \(x\) and \(y\) that cover this triangular region.
   - The triangle has the following vertices:
     - \( O(0, 0) \)
     - \( A(3, 7) \)
     - \( B(5, 7) \)

   The line segments of the triangle include:
   - From \( O \) to \( A \): This line has the equation \( y = \frac{7}{3}x \).
   - From \( O \) to \( B \): This line has the equation \( y = \frac{7}{5}x \).
   - From \( A \) to \( B \): This is a horizontal line where \( y = 7 \).

2. **Setting up the double integral**:
   We need to integrate over \(x\) and \(y\), where \(x\) ranges from \(0\) to \(5\) (since the \(x\)-coordinates of the triangle range from 0 to 5). For a given \(x\), the \(y\)-bounds are determined by the lines:
   - Lower bound: \( y = 0 \)
   - Upper bound: Between \( y = \frac{7}{5}x \) and \( y = 7 \) depending on \(x\).

So, we split the integral into two parts based on where \(x\) lies:
   - For \( 0 \leq x \leq 3 \), \( y \) goes from \( 0 \) to \( \frac{7}{3}x \).
   - For \( 3 \leq x \leq 5 \), \( y \) goes from \( 0 \) to \( 7 \).

Thus, the double integral becomes:
\[
\int_0^3 \int_0^{\frac{7}{3}x} (4 - 8x) \, dy \, dx + \int_3^5 \int_0^7 (4 - 8x) \, dy \, dx
\]

3. **Evaluating the integrals**:

   - **First part** (\( 0 \leq x \leq 3 \)):

   \[
   \int_0^3 \left[ \int_0^{\frac{7}{3}x} (4 - 8x) \, dy \right] dx
   \]

   The inner integral with respect to \(y\) is straightforward:
   \[
   \int_0^{\frac{7}{3}x} (4 - 8x) \, dy = (4 - 8x) \cdot \frac{7}{3}x
   \]
   So the first part becomes:
   \[
   \int_0^3 \frac{7}{3}x (4 - 8x) \, dx
   \]

   - **Second part** (\( 3 \leq x \leq 5 \)):

   \[
   \int_3^5 \left[ \int_0^7 (4 - 8x) \, dy \right] dx
   \]

   The inner integral is:
   \[
   \int_0^7 (4 - 8x) \, dy = 7(4 - 8x)
   \]
   So the second part becomes:
   \[
   \int_3^5 7(4 - 8x) \, dx
   \]

4. **Performing the final calculations**:

   Now we can compute the integrals and sum them.

I will calculate the numerical values now.The value of the double integral is \( -518 \).

Would you like further clarification or details on any step? Here are five related questions to expand your understanding:

1. How would the result change if the function was \( f(x, y) = 4 - 8y \) instead?
2. What is the geometric interpretation of a double integral over a triangular region?
3. Can you calculate the area of the triangle defined by the vertices \( O = (0, 0) \), \( A = (3, 7) \), and \( B = (5, 7) \)?
4. How would the bounds of the integral change if the triangle had a vertex at \( (5, 10) \) instead of \( (5, 7) \)?
5. How do we convert this triangular region into polar coordinates for integration?

**Tip**: When working with double integrals over non-rectangular regions, carefully consider the bounds based on the shape of the region.

Calculate the double integral of f(x, y) = 4 - 8x over the triangle with vertices O = (0,0), A = (3,7), B = (5,7).

Learn how to calculate the double integral of the function f(x, y) = 4 - 8x over a triangle with vertices O = (0,0), A = (3,7), B = (5,7). This solution includes detailed steps to set up and evaluate the integral, covering key calculus and geometry concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation