Homework questions thus far??? Section 4.10? 5.1? 5.2?

Homework questions thus far???

Section 4.10? 5.1? 5.2?

sin 2x

sin xdx∫

The Definite Integral

Chapters 7.7, 5.2 & 5.3

January 30, 2007

Estimating Area vs Exact Area

Pictures

Riemann sum rectangles, ∆t = 4 and n = 1:

Better Approximations Trapezoid Rule uses straight lines

Trapezoidal Rule

Better Approximations The Trapezoid Rule uses small lines Next highest degree would be

parabolas…

Simpson’s RuleMmmm…parabolas…

Put a parabola across each pair of subintervals:

Simpson’s RuleMmmm…parabolas…

Put a parabola across each pair of subintervals:

So n must be even!

Simpson’s Rule Formula

Like trapezoidalrule


Divide by 3instead of 2


Interiorcoefficientsalternate:

4,2,4,2,…,4


Second from start and end

are both 4

Simpson’s Rule Uses Parabolas to fit the curve

f (x)dxa

b

∫ ≈Δx3[ f (x0 ) + 4 f (x1) + 2 f (x2 ) + 4 f (x3) + ...

+4 f (xn−1) + f (xn)]

Where n is even and ∆x = (b - a)/n

S2n=(Tn+ 2Mn)/3

Use Simpson’s Rule to Approximate the definite integral with n = 4

g(x) = ln[x]/x on the interval [3,11]

Use T4.

Runners: A radar gun was used

to record the speed of a runner during the first 5 seconds of a race (see table) Use Simpsons rule to estimate the distance the runner covered during those 5 seconds.

t(s) v(m/s)0 0

0.5 4.671 7.34

1.5 8.862 9.73

2.5 10.223 10.51

3.5 10.674 10.76

4.5 10.815 10.81

Definition of Definite Integral: If f is a continuous function defined for a≤x≤b, we

divide the interval [a,b] into n subintervals of equal width ∆x=(b-a)/n. We let x0(=a),x1,x2,…,xn(=b) be the endpoints of these subintervals and we let x1

*, x2*, …

xn* be any sample points in these subintervals so

xi*lies in the ith subinterval [xi-1,xi]. Then the Definite

Integral of f from a to b is:

f (x)dxa

b

∫ =limn→ ∞

f(xi* )

i=1

n

∑ Δx

Express the limit as a Definite Integral

limn→ ∞

e1+4 in

⎛⎝⎜

⎞⎠⎟

2 +4in

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟i=1

n

∑ 4n

limn→ ∞

7n2 3+

7in

⎛⎝⎜

⎞⎠⎟+ 3+

7in

⎛⎝⎜

⎞⎠⎟

2

i=1

n

∑

Express the Definite Integral as a limit

(2−x2 )dx0

2

∫

tan 2x( )dx1

5

∫

Properties of the Definite Integral



Properties of the Integral

1) f (x)dx =a

b

∫ − f(x)dxb

a

∫

f (x)dxa

a

∫

cf (x)dx =c f(x)dxa

b

∫a

b

∫

2) = 0

3) for “c” a constant


Given that:

2 f (x)dx =8−2

1

∫

f(x)dx=31

4

∫

g(x)dx=2

−2

∫ 5

g(x)dx=2

4

∫ −7

Evaluate the following:

f (x)dx4

1

∫ =?

f (x)dx−2

1

∫ =?

3dx−1

1

∫ =?


Given that:

2 f (x)dx =8−2

1

∫

f(x)dx=31

4

∫

g(x)dx=2

−2

∫ 5

g(x)dx=2

4

∫ −7

Evaluate the following:

3g(x)dx−2

2

∫ =?

[3 f (x)−2g(x)]dx−2

4

∫ =?

Given the graph of f, find:

f (x)dx−1

4

∫

Evaluate:

f (x)dx−1

3

∫ f(x) =1−x2 −1≤x≤01 0 ≤x≤12 −x 1≤x≤3

⎧

⎨⎪

⎩⎪

Integral Defined Functions

Let f be continuous. Pick a constant a.

Define: F(x)= f(t)dta

x

∫



Define:

Notes:

• lower limit a is a constant.

F(x)= f(t)dta

x

∫



Define:

Notes:

• lower limit a is a constant.• Variable is x: describes how far to integrate.

F(x)= f (t)dta

x

∫



Define:

Notes:

• lower limit a is a constant.• Variable is x: describes how far to integrate.• t is called a dummy variable; it’s a placeholder

F(x)= f (t)dta

x

∫



Define:

Notes:

• lower limit a is a constant.• Variable is x: describes how far to integrate.• t is called a dummy variable; it’s a placeholder• F describes how much area is under the curve up to x.

F(x)= f(t)dta

x

∫

Example

Let . Let a = 1, and .

Estimate F(2) and F(3).

F(x)= f (t)dta

x

∫f (x)= 2 + x

F(x)= 2 + tdt1

x

∫

F(2)= 2 + tdt1

2

∫ ≈1 / 2

3f (1)+ 4 f(1.5) + f(2)[ ] ≈

1.8692

Example

Let . Let a = 1, and .

Estimate F(2) and F(3).

F(x)= f (t)dta

x

∫f (x)= 2 + x

F(x)= 2 + tdt1

x

∫

F(3)= 2 + tdt1

3

∫≈

1 / 2

3f (1) + 4 f (1.5) + 2 f (2) + 4 f (2.5) + f (3)[ ]

≈1.8692

Where is increasing and decreasing?

is given by the graph below: f (t)F is increasing. (adding area)

F is decreasing.

(Subtracting area)

F(x)= f (t)dta

x

∫

Fundamental Theorem I

Derivatives of integrals:

Fundamental Theorem of Calculus, Version I:

If f is continuous on an interval, and a a number on that interval, then the function F(x) defined by

has derivative f(x); that is, F'(x) = f(x).

F(x)= f(t)dta

x

∫

Example

Suppose we define .F(x)= cos(t2 )dt2.5

x

∫

Example

Suppose we define .

Then F'(x) = cos(x2).

F(x)= cos(t2 )dt2.5

x

∫

Example

Suppose we define .

Then F'(x) =

F(x)= (t2 + 2t+1)dt−7

x

∫

Example

Suppose we define .

Then F'(x) = x2 + 2x + 1.

F(x)= (t2 + 2t+1)dt−7

x

∫

Examples:

d

dxsin(t)

−π

x

∫ dt⎛

⎝⎜⎞

⎠⎟

d

dy5x2dx

y

2

∫⎛

⎝⎜

⎞

⎠⎟ =

d

dy− 5x2dx

2

y

∫⎛

⎝⎜

⎞

⎠⎟

d

drcos t dt

−π

r

∫

Examples:

d

dθx2dx

0

θ 3

∫⎛

⎝⎜

⎞

⎠⎟

d

drtan t 3 dt

−π

2r

∫d

drtan t 3 dt

−π

2r

∫

d

dθx2dx

0

θ 3

∫⎛

⎝⎜

⎞

⎠⎟

d

dxF[g(x)][ ] =F '[g(x)]g'(x)

If f is continuous on [a, b], then the function defined by

is continuous on [a, b] and differentiable on (a, b) and

Fundamental Theorem of Calculus (Part 1)

F(x)= f(t)dta

x

∫ a≤x≤b

F '(x)= f(x)

If f is continuous on [a, b], then the function defined by

is continuous on [a, b] and differentiable on (a, b) and

Fundamental Theorem of Calculus (Part 1)(Chain Rule)

F(x)= f(t)dta

u(x)

∫ a≤x≤b

F '(x)= f(u(x))u'(x)

In-class Assignment

a. Estimate (by counting the squares) the total area between f(x) and the x-axis.

b. Using the given graph, estimatec. Why are your answers in parts (a) and (b) different?

€

f (x)dx0

8

∫

d

dxln t dt1

2

cos x

∫⎛

⎝⎜⎞

⎠⎟

2.

1. Find:

First let the bottom bound = 1, if x >1, we calculate the area using the formula for trapezoids:

1

2b1 +b2( ) h( )

Consider the function f(x) = x+1 on the interval [0,3]

Now calculate with bottom bound = 1, and x < 1, :



So, on [0,3], we have that

And F’(x) = x + 1 = f(x) as the theorem claimed!

Very Powerful!Every continuous function is the derivative of some

other function! Namely:

F(x)=12x2 + 2x−3( )

f (t)dta

x

∫

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Homework questions thus far??? Section 4.10? 5.1? 5.2?