lets take a trip back in time…to geometry. can you find the area of the following? if so, why?

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Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

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Page 1: Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

Page 2: Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

Now, lets take a trip back to Advanced Algebra. Can you find the area of the region bounded by the line x=0, y=0 , y = 4 and y = 2x+3? If so, how?

3

04

Page 3: Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

04

16

Page 4: Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

When we find the area under a curve by adding rectangles, the answer is called a Riemann Sum.

subinterval

partition

The width of a rectangle is called a subinterval.

The entire interval is called the partition.

Subintervals do not all have to be the same size.

Page 5: Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

subinterval

partition

If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by .P

As gets smaller, the approximation for the area gets better. Why?

P

0

1

Area limn

k kP

k

f c x

if P is a partition of the interval ,a b

Page 6: Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

0

1

limn

k kP

k

f c x

is called the definite integral of

over .f ,a b

If we use subintervals of equal length, then the length of a

subinterval is:b a

xn

The definite integral is then given by:

1

limn

kn

k

f c x

Page 7: Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

1

limn

kn

k

f c x

Leibnitz introduced a simpler notation for the definite integral:

1

limn b

k ank

f c x f x dx

Note that the very small change in x becomes dx.

Page 8: Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

b

af x dx

IntegrationSymbol

lower limit of integration

upper limit of integration

integrandvariable of integration

(dummy variable)

It is called a dummy variable because the answer does not depend on the variable chosen.

Page 9: Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

b

af x dx

We have the notation for integration, but we still need to learn how to evaluate the integral.

This will be another day.

We will master the Riemann Sum work first! Onwards!!!

Page 10: Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

time

velocity

After 4 seconds, the object has gone 12 feet. Why?

Consider an object moving at a constant rate of 3 ft/sec.

Since rate . time = distance:

If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.

ft3 4 sec 12 ft

sec

3t d d’ ?

Page 11: Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

If the velocity is not constant,we might guess that the distance traveled is still equalto the area under the curve.

(The units work out.)

211

8V t Example:

We could estimate the area under the curve by drawing rectangles touching at their left corners.

We call this the Left-hand Rectangular Approx. Method (LRAM).

Approx. area:

Page 12: Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

We could also use a Right-hand Rectangular Approximation Method(RRAM).

Approx. area:

211

8V t

Page 13: Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method (MRAM).

1.031251.28125

1.78125Approx area: 2.53125

In this example there are four subintervals. As the number of subintervals increases, so does the accuracy.

211

8V t

Page 14: Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

211

8V t

Approximate area:6.65624

t v

1.007810.25

0.75 1.07031

1.25 1.19531

1.382811.75

2.25

2.75

3.25

3.75

1.63281

1.94531

2.32031

2.75781

13.31248 0.5 6.65624

width of subinterval

With 8 subintervals:

The exact answer for thisproblem is .6.6

Page 15: Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

Circumscribed rectangles are all above the curve:

Inscribed rectangles are all below the curve:

What Riemann Method?

Over or under estimate?

Concave up or down?

What Riemann Method?

Over or under estimate?

Concave up or down?

Page 16: Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

Riemann Sums Exercise Handout