Let's solve the problem step by step: ### **Step 1: Understanding the Problem** We are given a table of values of \( f(x, y) \), where \( f(x, y) \) is the number of milligrams of mosquito larvae per square meter in a swamp. - \( x \) values: 0, 5, 10 (meters) - \( y \) values: 0, 3, 6 (meters) - The region \( R \) is the rectangle \( 0 \leq x \leq 10 \), \( 0 \leq y \leq 6 \). We are asked to **estimate** the double integral: \[ \iint_R f(x, y) dA \] This represents the total number of milligrams of larvae in the entire rectangle. --- ### **Step 2: Approximate the Integral (Using Midpoint or Grid)** Since we have a grid of values, we can use the **midpoint rule** or **rectangular rule** (also like the double Riemann sum). The spacing: - In \( x \): \( 5 \) meters (from 0 to 5, 5 to 10) - In \( y \): \( 3 \) meters (from 0 to 3, 3 to 6) Each small rectangle is \( 5 \times 3 = 15 \) m². --- ### **Step 3: Sum the Function Values** There are \( 3 \times 3 = 9 \) grid points. Let's sum the values: \[ \begin{array}{c|ccc} & x=0 & x=5 & x=10 \\ \hline y=0 & 1 & 2 & 4 \\ y=3 & 2 & 5 & 8 \\ y=6 & 5 & 9 & 14 \\ \end{array} \] Sum of all values: \[ 1 + 2 + 4 + 2 + 5 + 8 + 5 + 9 + 14 = 50 \] --- ### **Step 4: Multiply by the Area per Rectangle** Each value represents a \( 5 \) by \( 3 \) square: \( 15 \) m². \[ \text{Approximate Integral} \approx \text{sum of values} \times \text{area per rectangle} \] \[ = 50 \times 15 = 750 \] --- ### **Step 5: Final Answer** \[ \boxed{750 \text{ mg}} \] --- ### **Summary Table** | Step | Action | Result | |------------------|-------------------------------|------------| | Sum all table values | \( 1+2+4+2+5+8+5+9+14 \) | 50 | | Area per rectangle | \( 5 \times 3 \) | 15 m² | | Multiply sum by area | \( 50 \times 15 \) | 750 mg | --- ## **Final Answer** \[ \boxed{750} \] The estimated total mass of mosquito larvae in the swamp rectangle is **750 mg**.

Let's go through the problem carefully, step-by-step, with detailed calculations. --- ### **Given Data:** | \( y \backslash x \) | 0 | 5 | 10 | |----------------------|-----|-----|-----| | 0 | 1 | 2 | 4 | | 3 | 2 | 5 | 8 | | 6 | 5 | 9 | 14 | - The function \( f(z, y) \) represents milligrams of mosquito larvae per square meter. - The rectangle \( R \) spans: - \( 0 \leq z \leq 10 \) - \( 0 \leq y \leq 6 \) - The grid points are at: - \( z = 0, 5, 10 \) - \( y = 0, 3, 6 \) --- ### **Step 1: Identify the grid spacing** - \( \Delta z = 5 \) meters (between 0 and 5, and 5 and 10) - \( \Delta y = 3 \) meters (between 0 and 3, and 3 and 6) Each small rectangle (subregion) has an area: \[ \Delta A = \Delta z \times \Delta y = 5 \times 3 = 15 \, \text{m}^2 \] --- ### **Step 2: Understanding the approximation method** Since values are given at grid points, and the problem asks for an estimate, **using the average value of the function at the grid points and multiplying by the total area** is a reasonable approach (similar to a double Riemann sum). --- ### **Step 3: Sum of the function values** The grid points and their values: | \( y \backslash z \) | 0 | 5 | 10 | |----------------------|---|---|----| | 0 | 1 | 2 | 4 | | 3 | 2 | 5 | 8 | | 6 | 5 | 9 | 14 | Sum all these values: \[ 1 + 2 + 4 + 2 + 5 + 8 + 5 + 9 + 14 \] Calculate step-by-step: - \( 1 + 2 = 3 \) - \( 3 + 4 = 7 \) - \( 7 + 2 = 9 \) - \( 9 + 5 = 14 \) - \( 14 + 8 = 22 \) - \( 22 + 5 = 27 \) - \( 27 + 9 = 36 \) - \( 36 + 14 = 50 \) **Total sum of values:** \[ \boxed{50} \] --- ### **Step 4: Approximate the integral** The double integral estimate: \[ \iint_{R} f(z, y) \, dA \approx \sum_{i,j} f(z_i, y_j) \times \Delta A \] where the sum of all function values is 50. Thus, the total estimated number of milligrams: \[ \boxed{ \text{Total} \approx 50 \times 15 = 750 \, \text{mg} } \] --- ### **Final Answer:** \[ \boxed{\boxed{750 \text{ mg}}} \] This is the estimated total mass of mosquito larvae in the specified swamp region.