Math 3298 - Hampton - Lab 6 - Due 10/25

Lab reports should conform to the guidelines explained on the handout.  Collaboration during the lab is strongly encouraged, although the write-up of the lab report must be done individually.

Fubini's Theorem

We will examine Fubini's Theorem (on page 1027 in our text) in a little more detail in this lab while using the powerful commands Integrate and NIntegrate.  Fubini's theorem states that we can exchange the order of integration if the function is continuous on the domain of integration.  

Here are some examples of Integrate and NIntegrate in use.  Integrate will attempt to provide exact answers to indefinite or definite integrals, while NIntegrate will find a good numerical answer for definite integrals.  Note that Integrate leaves off all constants of integration - it only finds _a_ antiderivative, not all of them.

Integrate[x,x]

Integrate[x+1,x]

Integrate[x,{x,0,1}]

NIntegrate[x,{x,0,1}]

Integrate[1,x,y]

Integrate[x^4y^6Exp[x y],x,y]

Integrate[y Sin[x]/x,{x,0,3 Pi},{y,0,1}]

NIntegrate[y Sin[x]/x,{x,0,3 Pi},{y,0,1}]

??Integrate

??NIntegrate

Exercise

1.  Try to evaluate the integral of f[x_,y_]:=(x-y)/(x+y)^3 over the unit square 0 ≤ x ≤ 1,  0 ≤ y ≤ 1 using different orders of integration.  
	a) Do your answers make sense?  
	b) How does this relate to Fubini's theorem? 
	c) Use Plot3D and ContourPlot to describe the behavior of this function on the domain.  
	d) What do you feel is the correct value of this integral? Why?