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

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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

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