Question

If \(f\)is a continuous function, \(f(x,y) = k\), and \(R = (a,b) \times (c,d)\), then show that

Short answer

proved below

Step-by-step solution

If \(f(x,y) = k\) and, \(R = (a,b) \times (c,d)\), then

\(\begin{aligned}{l}\int\limits_a^b {\int\limits_c^d {KdA} } = \int\limits_a^b {\int\limits_c^d {kdA} } \\ = \int\limits_a^b {K\left( {\int\limits_c^d {dy} } \right)dx} \\ = \int\limits_a^b {k(y)_c^ddx} \\ = \int\limits_a^b {k(d - c)dx} \\ = k(d - c)\int\limits_a^b {dx} \Rightarrow k(d - c)(x)_a^b\\ = k(d - c)(b - a)\end{aligned}\)

Hence proved, that .