Matthew Ebert
Mat 329 Chapter 15.1 Problem Number 15

Clear [f,i,j,m,n]
f[x_, y_] := Sqrt[1 + x*E^-(y)]
N[Integrate[f[x, y], {x, 0, 1}, {y, 0, 1}], 9]
middlesum := 1/(m*n)*Sum[f[((i/m) + ((i - 1))/m)/2, ((j/n) + ((j - 1))/n)/2], {i, 1, m}, {j, 1, n}]
m := 1; n := 1
Evaluate[N[middlesum]]
m := 2; n := 2
Evaluate[N[middlesum, 9]]
m := 4; n := 4
Evaluate[N[middlesum, 9]]
m := 8; n := 8
Evaluate[N[middlesum, 9]]
m := 16; n := 16
Evaluate[N[middlesum, 9]]
m := 32; n := 32
Evaluate[N[middlesum, 9]]