suppose $\\int_{0}^{2} f(x) d x=-1$ and $\\int_{4}^{7} g(x) d x=-2$. determine the value of $\\iint_{d} f(x)…

Explanation:

Step1: Express double - integral as an iterated integral

By the property of double - integral over a rectangular region (D = [a,b]\times[c,d]), (\iint_{D}f(x)g(y)dA=\int_{a}^{b}\int_{c}^{d}f(x)g(y)dydx). For (D = [0,2]\times[4,7]), we have (\iint_{D}f(x)g(y)dA=\int_{0}^{2}\left(\int_{4}^{7}f(x)g(y)dy\right)dx). Since (f(x)) is independent of (y) and (g(y)) is independent of (x), we can rewrite it as (\int_{0}^{2}f(x)dx\int_{4}^{7}g(y)dy). (Using the property (\int_{a}^{b}k\cdot h(t)dt = k\int_{a}^{b}h(t)dt) where (k) is a constant with respect to (t))

Step2: Substitute the given integral values

We know that (\int_{0}^{2}f(x)dx=- 1) and (\int_{4}^{7}g(x)dx = - 2). Since (\int_{4}^{7}g(y)dy=\int_{4}^{7}g(x)dx) (the variable of integration is a dummy variable). Then (\int_{0}^{2}f(x)dx\int_{4}^{7}g(y)dy=(-1)\times(-2))

Answer:

(2)