Final answer:
To evaluate the given double integral, we need to integrate the function 6cos(x,y) with respect to x and y within the given limits. By performing the integration step by step, we arrive at the result 12sin(7) + 2C(y), where C(y) is a constant that depends on y.
Step-by-step explanation:
To evaluate the double integral ∬6cos(x,y)dxdy, where 0≤x≤7 and 0≤y≤2, we need to integrate the function 6cos(x,y) with respect to both x and y within the given limits. Let's start by integrating with respect to x first.
Integrating 6cos(x,y) with respect to x, we get 6sin(x,y) + C(y), where C(y) is the constant of integration which depends on y.
Next, we integrate 6sin(x,y) + C(y) with respect to y from 0 to 2. Since C(y) is independent of y in this case, it can be treated as a constant.
Integrating 6sin(x,y) with respect to y, we get 6y*sin(x,y) + C(y)*y. Evaluating this expression from 0 to 2, we get 12sin(x,y) + 2C(y).
Thus, the double integral evaluates to 12sin(7) + 2C(y), where C(y) is a constant that depends on y.