Final Answer:
The double integral of f(x, y) = 7a + 5y over the region R bounded by y=0, y=2√x, and x=d ², where d=2 is approximately 96.00.
Step-by-step explanation:
To solve this double integral, we first need to set up the limits of integration. The given region R is defined by y=0, y=2√x, and x=d ². We are also given that d=2. Substituting this into the expressions for y, we get the limits for the integral as y=0 to y=4 and x=4 to x=16.
The integral becomes ∫ from 4 to 16 ∫ from 0 to 4 of (7a + 5y) dy dx. First, integrate with respect to y, treating x as a constant: ∫ from 4 to 16 [7ay + (5/2)y²] evaluated from y=0 to y=4 dx. Simplifying this gives ∫ from 4 to 16 (28a + 40) dx.
Now, integrate with respect to x: [28ax + 40x] evaluated from x=4 to x=16. Substituting these values gives (448a + 640) - (112a + 160), which simplifies to 336a + 480. Since a is not given, we can't find the exact numerical value. However, for a general case, let a=1, yielding the result of 816.
To find the approximate value, substitute a=1 and evaluate the expression, giving 816.00. Rounding to two decimal places, the final answer is approximately 96.00.