The Fundamental Theorem of Calculus

2 December 2008

Katie Ford

Math 101

So what’s the deal?

The Fundamental Theorem of Calculus (FTC) links the two branches of calculus: differentiation and integration.

This seems completely obvious to us now, but it really wasn’t originally.

History

Isaac Barrow (1630-1677)

Isaac Newton (1642-1726)

Gottfried Leibnitz (1646-1716)

Definitions

Let f: D→R. f is uniformly continuous on D if for every ε>0 there exists a δ>0 such that |f (x)-f (y)| < ε whenever |x-y| < δ and x,y є D.

Let [a,b] be an interval in R. A partition P of [a,b] is a finite set of points {x0, x1,…, xn} in [a,b] such that a = x0 <x1 <…<xn= b

More Definitions: Upper and Lower Sums and Integrals

Upper and Lower Sum:

Suppose that f is a bounded function defined on [a,b] and that P= {x0,…,xn} is a partition of [a,b]. For each i=1,…n, we let

Mi(f)=sup{f(x): x є [xi-1, xi]}

and

mi(f)= inf {f(x): x є [xi-1, xi]}

We let xi=xi-xi-1 (i= 1,…,n), and then the upper and lower sums of f is defined with respect to P to be:

Upper integral- U(f)- inf {U(f,P): P is a partition of [a,b]} Lower integral- L(f)- sup {L(f,P): P is a partition of [a,b]} Integrable- if U(f)=L(f)=∫a

bf(x)dx

Theorems dealing with properties of Integrals

kf is integrable and ∫abkf = k∫a

bf. Also f+g is integrable and ∫a

b(f+g) = ∫abf + ∫a

bg.

If f is integrable on both [a,c] and [c,b], then f is integrable on [a,b]. Furthermore, ∫a

bf =∫acf +∫c

bf

The FIRST Fundamental Theorem of Calculus

Let f be integrable on [a,b]. For each x є [a,b], let

Then F is uniformly continuous on [a,b]. Furthermore, if f is continuous at c є [a,b], then F is differentiable at c and

Proof:

Since f is integrable on [a,b], it is bounded there. That is, there exists B>0 such that |f (x)| ≤ B for all x є [a,b]. To see that F is uniformly continuous on [a,b], let ε > 0 be given. If x,y є [a,b] with x<y and |x-y|< ε/B, then

Thus F is uniformly continuous on [a,b].

Now suppose that f is continuous at c є [a,b]. Then given any ε >0, there exists a δ>0 such that |f(t)-f(c)|< ε whenever t є [a,b] and |t-c|< δ. Since f(c) is a constant, we may write:

Proof (continued)

Then for any x є [a,b] with 0<|x-c|<δ, we have:

Since >0 was arbitrary, we conclude that:

The SECOND Fundamental Theorem of Calculus

If f is differentiable on [a,b] and f’ is integrable on [a,b], then:

Proof:

Let P = {x0, x1,…, xn} be any partition of [a,b]. We apply the Mean Value Theorem to each subinterval [xi-1,xi] and obtain points ti є (xi-1,xi) such that:

Since mi(f ’) ≤ f ’(ti) ≤ Mi(f ’) for all i, it follows that L(f ’,P) ≤ f(b)-f(a) ≤ U(f ’,P)

Since this holds for each partition P, we also have L(f’) ≤ f(b)-f(a) ≤ U(f ’)

But f’ is assumed to be integrable on [a,b], so L(f ’) = U(f ’) = ∫ab f ’

Thus, f(b)-f(a)=∫ab f ’

Bibliography

Lay, Steven R. Analysis With an Introduction to Proof. 4th ed. Upper Saddle River, NJ: Pearson: 2005.

Bardi, Jason Socrates. The Calculus Wars. New York: Thunder’s Mouth Press, 2006.

Stewart, James. Calculus, Concepts and Contexts. 2nd ed. Pacific Grove, CA: Wadsworth Group, 2001.