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Calculus for Biologists

The Fundamental Theorem of Calculus

James K. Peterson

Department of Biological Sciences and Department of Mathematical SciencesClemson University

August 29, 2013


Outline

1 Properties of The Riemann Integral

2 Fundamental Theorem Of Calculus

3 The Cauchy Fundamental Theorem Of CalculusExamples


Abstract

This lecture explains the amazing connection between the idea ofan antiderivative or primitive of a function and the Riemannintegral of that same function on some interval.


Properties of The Riemann Integral

If you think about how the Riemann integral is defined in terms oflimits of Riemann sums, it is pretty easy to figure out some basicproperties.∫ a

a f (t)dt = 0 as the partitions are all one point and all the

changes in t are 0. Thus,∫ 11 t2dt = 0 as the interval is just

one point.∫ ba f (t)dt =

∫ ca f (t)dt +

∫ bc f (t)dt for any c between a and

b. For example:∫ 5

1t2dt =

∫ 3

1t2dt +

∫ 5

3t2dt.∫ 5

1t2dt =

∫ 2.2

1t2dt +

∫ 5

2.2t2dt

Having the order backwards just changes the sign of theRiemann integral value. So∫ 1

5t2dt = −

∫ 5

1t2dt.∫ −1

8t3dt = −

∫ 8

−1t3dt


Fundamental Theorem Of Calculus

There is a big connection between the idea of the antiderivativeof a function f and its Riemann integral.

For a positive function f on the finite interval [a, b], we canconstruct the area under the curve functionF (x) =

∫ xa f (t) dt.

Let’s look at the difference in these areas: we assume h ispositive.

F (x + h) − F (x) =

∫ x+h

af (t) dt −

∫ x

af (t) dt

=

∫ x

af (t) dt +

∫ x+h

xf (t) dt

−∫ x

af (t) dt

where we have used standard properties of the Riemannintegral to write the first integral as two pieces.



Now subtract to get

F (x + h) − F (x) =

∫ x+h

xf (t) dt

Now divide this difference by the change in x which is h. Wefind

F (x + h) − F (x)

h=

1

h

∫ x+h

xf (t) dt

We show F (x) and F (x + h) for a small positive h in thefigure which follows.



F (x) is the area under this curve from a to x .

(a, f (a))(b, f (b))

a bx x + h

F (x) F (x + h)

Figure: The Function F (x)



The difference in area,∫ x+hx f (t) dt, is the second shaded area in

the figure you just looked at. We see

If t is any number between x and x + h, the area of therectangle with base h and height f (t) is f (t) × h which isclosely related to the area difference.

Note the difference between this area and F (x + h) − F (x)is really small when h is small.

We know that f is bounded on [a, b] You can easily see that fhas a maximum value for the particular f we draw. Of course,this graph is not what all such bounded functions f look like,but you should be able to get the idea that there is a numberB so that 0 < f (t) ≤ B for all t in [a, b].

Thus, we see

F (x + h) − F (x) ≤∫ x+h

xB dt = B h (1)



From this, it follows that

We see our difference lives between 0 and B.

0 ≤ (F (x + h) − F (x)) ≤ B h

And so taking the limit as h gets small, we find

0 ≤ limh→ 0

(F (x + h) − F (x))

≤ limh→ 0

B h = 0.

We conclude that F is continuous at each x in [a, b] as

limh→ 0

(F (x + h) − F (x)) = 0.

It seems that the new function F we construct by integratingthe function f in this manner always builds a new functionthat is continuous.



Is F differentiable at x? Let’s do an estimate. We have a lowerand upper bound on the area of the middle slice in our figure.

minx ≤t ≤x+h

f (t) h ≤∫ x+h

xf (t)dt ≤ max

x ≤t ≤x+hf (t) h

Thus, we have the estimate

minx ≤t ≤x+h

f (t) ≤ F (x + h) − F (x)

h≤ max

x ≤t ≤x+hf (t)



If f was continuous at x , then we must have

limh→ 0

minx ≤t ≤x+h

f (t) = f (x)

and

limh→ 0

maxx ≤t ≤x+h

f (t) = f (x)

Note the f we draw in our figure is continuous all the time,but the argument we use here only needs continuity at thepoint x! At any rate, we can infer for positive h,

limh→ 0+

F (x + h) − F (x)

h= f (x)



You should be able to believe that a similar argument wouldwork for negative values of h: i.e.,

limh→ 0−

F (x + h) − F (x)

h= lim

h→ 0−f (t) = f (x)

This tells us that F ′(p) exists and equals f (x) as long as f iscontinuous at x as

F ′(x+) = limh→ 0+

F (x + h) − F (x)

h= f (x)

F ′(x−) = limh→ 0−

F (x + h) − F (x)

h= f (x)



This relationship is called the Fundamental Theorem ofCalculus.

Our argument works for x equals a or b but we only need tolook at the derivative from one side. So the discussion is a bitsimpler.

Our argument used a positive f but it works just fine if f haspositive and negative spots. Just divide f into it’s postive andnegative pieces and apply these ideas to each piece and thenglue the result together.

We can actually prove this using fairly relaxed assumptions onf for the interval [a, b]. In general, f need only be RiemannIntegrable on [a, b] which allows for jumps in the function.But those arguments are more advanced!



Theorem

The Fundemental Theorem of CalculusLet f be Riemann Integrable on [a, b]. Then the function Fdefined on [a, b] by F (x) =

∫ xa f (t) dt satisfies

1 F is continuous on all of [a, b]

2 F is differentiable at each point x in [a, b] where f iscontinuous and F ′(x) = f (x).



We can do more!!!

Using the same f as before, suppose G was defined on [a, b]as follows

G (x) =

∫ b

xf (t) dt.

then

F (x) + G (x) =

∫ x

af (t) dt +

∫ b

xf (t) dt

=

∫ b

af (t) dt.



So

G (x) =

∫ b

af (t) dt − F (x)

Since the Fundamental Theorem of Calculus tells us F isdifferentiable, we see G (x) must also be differentiable. Itfollows that since the derivative of a constant is 0, we have

G ′(x) = − F ′(x) = −f (x).



Let’s state this as a variant of the Fundamental Theorem ofCalculus, the Reversed Fundamental Theorem of Calculus soto speak.

Theorem

Reversed Fundamental Theorem of CalculusLet f be Riemann Integrable on [a, b]. Then the function G

defined on [a, b] by G (x) =∫ bx f (t) dt satisfies

1 G is continuous on all of [a, b]

2 G is differentiable at each point x in [a, b] where f iscontinuous and G ′(x) = −f (x).


The Cauchy Fundamental Theorem Of Calculus

We can use the Fundamental Theorem of Calculus to learn how toevaluate many Riemann integrals. This is how it works.

If f is continuous, the FToC tells us that F (x) =∫ xa f (t)dt

satisfies F ′(x) = f (x). So F is an antiderivative of f !!

If G is another antiderivative of f , then it also satisfiesG ′(x) = f (x).

We can use this to figure out a way to evaluate Riemannintegrals!



Here’s the argument (very cool one too I might add!)

For our f which is continuous on [a, b], let

F (x) =

∫ x

af (t) dt,

So F ′ = f and note F (a) = 0.

Let G be an antiderivative of the function f on [a, b]. Then,by definition, G ′(x) = f (x) and also G is continuous since itis differentiable.

Let H = F − G . Then

H ′(x) =

(F (x) − G (x)

)′= f (x) − f (x) = 0.

The only function whose derivative is 0 is a constant. So forsome constant C ,

H(x) = F (x) − G (x) = C



Almost done!

Thus H(a) = H(b) = C as H has the same value everywhere.

But H(a) = F (a) − G (a) and H(b) = F (b) − G (b).

These values are the same, so

F (a) − G (a) = F (b) − G (b).

Rearranging, we have

F (b) =

∫ b

af (t)dt = G (b) − G (a).

This result is huge! It says we can evaluate any Riemannintegral if we can guess an antiderivative.



Let’s formalize this as a theorem called the Cauchy FundamentalTheorem of Calculus. All we really need to prove this result isthat f is Riemann integrable on [a, b], which for us is usually trueas our functions f are continuous in general.

Theorem

Cauchy Fundamental Theorem of CalculusLet f be Riemann integral on [a, b] and let G be any

antiderivative of f . Then G (b) − G (a) =∫ ba f (t) dt.

We usually write this as∫ b

af (t)dt = G (t)

∣∣∣∣ba



Examples

In the problems that follow it doesn’t matter which antiderivativewe choose as our result above doesn’t care. So we just chooseC = 0 always.

∫ 3

1t3 dt =

t4

4

∣∣∣∣31

=34

4− 14

4=

80

4

∫ 4

−2t3 dt =

t4

4

∣∣∣∣4−2

=44

4− (−2)4

4

=256

4− 16

4=

240

4

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