# Calculate $$\int_R\int f(x,y) dA$$ for f(x,y) = 100 – 6x2y and $$R: 0 \leq x \leq 2, -1 \leq y \leq 1$$

Solution

Given that

f(x,y) = 100 – 6x2y and $$R: 0 \leq x \leq 2, -1 \leq y \leq 1$$

Now,

$$= \int_R\int f(x,y) dA$$

$$= \int_{-1}^1\int_{0}^2 (100 – 6x^2y) dx dy$$

$$= \int_{-1}^1 \left [ 100x – 2x^3y \right ]_0^2 \enspace dy$$

$$= \int_{-1}^1 \left [ 200 – 16y \right ] \enspace dy$$

$$= \left [ 200 – 16y \right ]_{-1}^1$$

= (200 – 8) – (-200 – 8)

= 192 + 208

= 400

Thus, $$\int_R\int f(x,y) dA$$ = 400