(865, #11) Use the Midpoint Rule for double integrals (see Section 13.1) with four squares to estimate the surface area of the portion of the paraboloid z = x² + y² that lies above the square [0,1] x [0,1]

Solution: The surface is in green, the domain of integration in yellow:

The integrand is sqrt(1 + 4 x² + 4 y²)

To estimate using the midpoint rule we first need to find the midpoints of the 4 squares.

The midpoints are (1/4,1/4), (3/4,1/4), (1/4,3/4), (3/4,3/4). Thus the mid-point approximation for the integral is

Here is how to do the problem with Maple:

g := (x,y) -> sqrt(1+4*x^2 + 4*y^2):

m := 2: n := 2:
total :=sum( sum( g( (i+0.5)/m, (j+0.5)/n )*(1/m)*(1/n), i = 0..m-1), j = 0..n-1):
evalf(total);

1.827902534

by: gm