Use the Midpoint Rule to estimate the volume under \(f\left( {x,y} \right) = 4x + 8y\) and above the rectangle given by \(0 \le x \le 4\), \(2 \le y \le 6\) in the \(xy\)-plane. Use 4 subdivisions in the \(x\) direction and 4 subdivisions in the \(y\) direction.
Use the Midpoint Rule to estimate the volume under \(f\left( {x,y} \right) = 4x - {y^2}\) and above the rectangle given by \(0 \le x \le 2\), \( - 2 \le y \le 1\) in the \(xy\)-plane. Use 2 subdivisions in the \(x\) direction and 3 subdivisions in the \(y\) direction.