Double Integrals over Rectangles: Examples

1. Use the Midpoint rule with m=n=2 to estimate the value of the integral x-3*y^2, over the region x=0..2, y=1..2

Anwser:

This can be done in one step with Maple using the ApproximateInt command. But we first need to activate the Multivariable package:

>	with(Student[MultivariateCalculus]);

>	ApproximateInt(x-3*y^2,x=0..2,y=1..2,partition=[2,2],method=midpoint);

We can also see a plot and animation using the same command:

>	ApproximateInt(x-3*y^2,x=0..2,y=1..2,partition=[2,2],method=midpoint,output=plot);

>	ApproximateInt(x-3*y^2,x=0..2,y=1..2,partition=[2,2],method=midpoint,output=animation);

2. Using your anwser from the last equation, find the average value of the function over the region specified above:

Anwser:

Divide by the area of the region:

>	-11.87500000/2;