Improper Integration

MAT 229, Spring 2021

Week 7

Supporting materials

If you wish to get a different perspective on the notes below, try either of the following textbook sections.

Stewart’s Calculus
	Section 7.8: improper Integrals

Boelkins/Austin/Schlicker’s Active Calculus
	Section 6.5: Improper Integrals

Improper integrals

An improper integral is a definite integral where either one or both of the limits is ∞, or the integrand is not defined for some value(s) of x between the limits of integration. Frequently there is a vertical asymptote causing trouble.

Proper integral: (∫_1^10)(1/(x^2))d x

Improper integral: (∫_1^∞)(1/(x^2))d x

Integrate[1/x^2,{x,1,Infinity}]

Improper integral: (∫_0^10)(1/(x^2))d x

Integrate[1/x^2,{x,0,10}]

We make sense out of an improper integral by turning it into a limit.

Improper integral: 
	(∫_1^∞)(1/(x^2))d x=(lim_(R->∞))(∫_1^R)(1/(x^2))d x
	=(lim_(R->∞))(∫_1^R)(x^(-2))d x
	=(lim_(R->∞))(-(x^(-1))(Subscript[|, 1]^R))
	=(lim_(R->∞))(-(R^(-1))+(1^(-1)))
	=(lim_(R->∞))(-(1/R)+1)=0+1=1
This improper integral converges to 1.

Improper integral:
	(∫_0^10)(1/(x^2))d x=(lim_(r->(SuperPlus[0])))(∫_r^10)(1/(x^2))d x
	=(lim_(r->(SuperPlus[0])))(∫_r^10)(x^(-2))d x
	=(lim_(r->(SuperPlus[0])))(-(x^(-1))(Subscript[|, r]^10))
	=(lim_(r->(SuperPlus[0])))(-(10^(-1))+(r^(-1)))
	=(lim_(r->(SuperPlus[0])))(-(1/10)+(1/r))=-(1/10)+∞
This improper integral diverges to ∞.

Questions

What makes (∫_0^∞)(1/(1+(x^2)))d x an improper integral? Does it converge or not? (Video)

Integrate[1/(1+x^2),{x,0,Infinity}]

What makes (∫_0^1)(1/(Sqrt[x]))d x an improper integral? Does it converge or not? (Video)

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

What makes (∫_0^2)(1/(((x-1))^(2/3)))d x an improper integral? Does it converge or not? (Video)

Integrate[1/(x-1)^(2/3),{x,0,2}]

Questions

1. Let S be the region between the x-axis and y=(1/x) for x≥1.

Is the area of S finite or not? (Video)

Integrate[1/x,{x,1,Infinity}]

Rotate S about the x-axis to create a solid of revolution. Is the volume of this solid finite or not? (Video)

Integrate[Pi/x^2,{x,1,Infinity}]

Does this result seem mysterious to you? The area of a cross-section infinite, but the volume of revolution of that area finite?

2. Let’s investigate various powers of x:

Evaluate (∫_1^∞)(1/(x^3))d x, (∫_1^∞)(1/x)d x, and (∫_1^∞)(1/(x^(1/3)))d x. (Video)

Plot[{1/x^3,1/x,1/x^(1/3)},{x,1, 10}]
Integrate[1/x^3,{x,1,Infinity}]
Integrate[1/x,{x,1,Infinity}]
Integrate[1/x^(1/3),{x,1,Infinity}]

For which values of p does (∫_1^∞)(1/(x^p))d x converge and for which p does it diverge? (Video)

Evaluate (∫_0^1)(1/(x^3))d x, (∫_0^1)(1/x)d x, and (∫_0^1)(1/(x^(1/3)))d x. (Video)

Plot[{1/x^3,1/x,1/x^(1/3)},{x,.1, 2}]
Integrate[1/x^3,{x,0,1}]
Integrate[1/x,{x,0,1}]
Integrate[1/x^(1/3),{x,0,1}]

For which values of p does (∫_0^1)(1/(x^p))d x converge and for which p does it diverge? (Video)

3. It’s not just powers of x, of course:

Evaluate (∫_0^1)ln(x)d x, (∫_0^∞)(e^(-x)) d x.

Integrate[Log[x],{x,0,1}]
Integrate[E^(-x),{x,0,Infinity}]

The consequence of changing the power on x: n=1 is the borderland....

This first “Manipulate” command shows the consequences of changing the power on x in the integral  (∫_0^1)(1/(x^p))d x from n=0.01 to 4:

Manipulate[Plot[(1/(x^n)),{x,0.1,2},PlotRange->{0,2},Filling->Axis, FillingStyle ->RGBColor[1,1,0],PlotLabel-> Integrate[x^(-n),{x,0,2}]],{n,1/100,4}]

This second “Manipulate” command shows the consequences of changing the power on x in the improper integral   (∫_1^∞)(1/(x^p))d x from n=0.01 to 4:

Manipulate[Plot[(1/(x^n)),{x,1,10},PlotRange->{0,1},Filling->Axis, FillingStyle ->RGBColor[1,1,0],PlotLabel-> Integrate[x^(-n),{x,1,Infinity}]],{n,.1,4}]

Things change dramatically at n=1 in both cases. n=1 is a very special value -- neither improper integral,  (∫_1^∞)(1/x)d x or (∫_0^1)(1/(x^p))d x,  converges. For all other values of n, one or the other improper integral converges. What else is special about  (1/x)? It’s the only power without a power as an antiderivative: its antiderivative is ln(x)! 

And ln(x) blows down to -∞ at 0, and blows up to ∞ as x approaches ∞.... That’s a very special power, (1/x)!

Homework

IMath problems on improper integrals.