Section 7.8: Improper Integration

Review

To evaluate a definite integral for which we can’t find a “nice” antiderivative use numerical integration. Let (Subscript[x, k])=a+k(((b-a)/n)) for k=0,1,2,…,n.

LRR rule:
	(∫_a^b)f(x)d x≈((b-a)/n)(Underoverscript[Σ, (k=1), n])f((Subscript[x, k-1]))

RRR rule (notice the subtle difference):
	(∫_a^b)f(x)d x≈((b-a)/n)(Underoverscript[Σ, (k=1), n])f((Subscript[x, k]))

Midpoint rule:
	(∫_a^b)f(x)d x≈((b-a)/n)(Underoverscript[Σ, (k=1), n])f((((Subscript[x, k-1])+(Subscript[x, k]))/2)),  
	Error≤(((K(b-a))^3)/(24(n^2)))

Trapezoid rule: is really just the average of the LRR and RRR rules, which are easily calculated. Nonetheless, one formula for Trapezoidal is  (∫_a^b)f(x)d x≈((b-a)/(2n))(f((Subscript[x, 0]))+2(Underoverscript[Σ, (k=1), (n-1)])f((Subscript[x, k]))+f((Subscript[x, n]))),  
	Error≤(((K(b-a))^3)/(12(n^2)))

Simpson’s rule: (∫_a^b)f(x)d x
≈((b-a)/(3n))(f((Subscript[x, 0]))+4f((Subscript[x, 1]))+2f((Subscript[x, 2]))+4f((Subscript[x, 3]))+2f((Subscript[x, 4]))+…+4f((Subscript[x, n-1]))+f((Subscript[x, n])))
=((b-a)/(3n))(f((Subscript[x, 0]))+4(Underoverscript[Σ, (k=1), (n/2)])f((Subscript[x, 2k-1]))+2(Underoverscript[Σ, (k=1), (n/2)])f((Subscript[x, 2k]))),  
	Error≤(((L(b-a))^5)/(180(n^4)))

Here, K is an upper bound for |f''(x)| for a≤x≤b, and L is an upper bound for |(f^((4)))(x)| for a≤x≤b. Again, Simpson’s rule is better thought of as an average of Midpoint and Trapezoidal, but a weighted average:	 (Subscript[S, 2n])=((2(Subscript[M, n])+(Subscript[T, n]))/3)

Questions

Consider the definite integral (∫_(-1)^1)sin((x^2))d x. Let f(x)=sin((x^2)).

f[x_]=Sin[(x^2)]
Integrate[f[x],{x,-1,1}]

A plot of |f''(x)| for -1≤x≤1 is shown below. Use it to determine a value of n for which the trapezoid rule will give an approximation to this integral with error less than 0.001.

Plot[Abs[f''[x]],{x,-1,1}]

A plot of |(f^((4)))(x)| for -1≤x≤1 is shown below. Use it to determine a value of n for which Simpson’s rule will give an approximation to this integral with error less than 0.001.

Plot[Abs[f''''[x]],{x,-1,1}]

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.

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}]

Make sense out of an improper integral by turning it into a limit problem.

Improper integral: (∫_1^∞)(1/(x^2))d x=(lim_(R->∞))(∫_1^R)(1/(x^2))d x

Improper integral: (∫_0^10)(1/(x^2))d x=(lim_(r->(SuperPlus[0])))(∫_r^10)(1/(x^2))d x

Questions

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

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?

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

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

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

Questions

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

Is the area of S finite or not?

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?

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

Questions

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

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}]

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

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}]

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

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

Here’s a manipulate command, to illustrate the consequence of changing the power on x:

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