Evaluate the double integral ${\int }_0^2{\int }_0^3(4-x-y)dydx$ by first identifying it as the volume of a solid. What is the value of the integral?

• A. $10$
• B. $12$
• C. $6$
• D. $8$

Answer: (D)

Verified step by step guidance

1. Step 1: Recognize that the double integral represents the volume of a solid bounded by the function z = 4 - x - y over the region defined by 0 ≤ x ≤ 2 and 0 ≤ y ≤ 3. The integral computes the accumulation of the function's values over this region.
2. Step 2: Write the double integral explicitly as \( \int_{0}^{2} \int_{0}^{3} (4 - x - y) \, dy \, dx \). The order of integration indicates that we first integrate with respect to \( y \), treating \( x \) as a constant, and then integrate with respect to \( x \).
3. Step 3: Perform the inner integral with respect to \( y \). The integral \( \int_{0}^{3} (4 - x - y) \, dy \) involves splitting the terms: \( \int_{0}^{3} 4 \, dy - \int_{0}^{3} x \, dy - \int_{0}^{3} y \, dy \). Compute each term separately.
4. Step 4: Substitute the results of the inner integral into the outer integral. After evaluating \( \int_{0}^{3} (4 - x - y) \, dy \), you will have an expression in terms of \( x \). Integrate this expression with respect to \( x \) over the interval \( [0, 2] \).
5. Step 5: Simplify the final result of the outer integral to find the total volume. This will give the value of the double integral, which corresponds to the volume of the solid.