Evaluate the double integral ${\int }_0^2{\int }_0^1(2x+1+xy) dy dx$.

• A. $12$
• B. $8$
• C. $10$
• D. $6$

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the problem. The given double integral is \( \int_0^2 \int_0^1 (2x + 1 + xy) \,dy\,dx \). This means we are integrating the function \( 2x + 1 + xy \) first with respect to \( y \) (inner integral) and then with respect to \( x \) (outer integral).
2. Step 2: Perform the inner integral. To integrate \( 2x + 1 + xy \) with respect to \( y \), treat \( x \) as a constant. The integral becomes \( \int_0^1 (2x + 1 + xy) \,dy \). Break it into separate terms: \( \int_0^1 2x \,dy + \int_0^1 1 \,dy + \int_0^1 xy \,dy \).
3. Step 3: Compute each term of the inner integral. For \( \int_0^1 2x \,dy \), the result is \( 2x \cdot y \) evaluated from \( y = 0 \) to \( y = 1 \). For \( \int_0^1 1 \,dy \), the result is \( y \) evaluated from \( y = 0 \) to \( y = 1 \). For \( \int_0^1 xy \,dy \), the result is \( x \cdot \frac{y^2}{2} \) evaluated from \( y = 0 \) to \( y = 1 \).
4. Step 4: Combine the results of the inner integral. Substitute the limits of integration \( y = 0 \) and \( y = 1 \) into each term. This gives the simplified expression for the inner integral as a function of \( x \).
5. Step 5: Perform the outer integral. Take the result from the inner integral and integrate it with respect to \( x \) over the interval \( [0, 2] \). Break it into separate terms if necessary, and compute each term using standard integration techniques. Substitute the limits \( x = 0 \) and \( x = 2 \) to find the final value of the double integral.