Final answer:
To solve the given double integral, evaluate the integral with respect to the inner variable x and then the outer variable y. The final result is 6.17.
Step-by-step explanation:
To solve the given double integral, we first need to evaluate the integral with respect to the inner variable x and then the outer variable y. Let's start with the inner integral:
∫ [0 to 2] 3(x + 4y)² dx
Integrating this expression with respect to x, we get: 3/3 (x + 4y)³, evaluated from 0 to 2:
(1/3)[(2 + 4y)³ - (0 + 4y)³] = (1/3)(8 + 12y + 6y² + 8y + 4y² + y³) - (4y³) = (1/3)(8 + 20y + 10y² + y³)
Now, we can integrate this expression with respect to y:
∫ [0 to 1] (1/3)(8 + 20y + 10y² + y³) dy
Integrating this expression, we get: (8/3)y + (10/3)y² + (5/2)y³ + (1/12)y⁴, evaluated from 0 to 1:
(8/3) + (10/3) + (5/2) + (1/12) = 6.17