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Presentation Transcript

What you’ll learn about

- Distance Traveled
- Rectangular Approximation Method (RAM)
- Volume of a Sphere
- Cardiac Output

… and why

Learning about estimating with finite sums sets the foundation for understanding integral calculus.

Section 5.1 – Estimating with Finite Sums

- Distance Traveled at a Constant Velocity:

A train moves along a track at a steady rate of

75 mph from 2 pm to 5 pm. What is the total distance traveled by the train?

v(t)

75mph

TDT = Area under line

= 3(75)

= 225 miles

t

2

5

Section 5.1 – Estimating with Finite Sums

- Distance Traveled at Non-Constant Velocity:

v(t)

75

Total Distance Traveled =

Area of geometric figure

= (1/2)h(b1+b2)

= (1/2)75(3+8)

= 412.5 miles

t

2 5 8

Example Estimating Area Under the Graph of a Nonnegative Function

Applying LRAM on a graphing calculator using 1000 subintervals, we find the left endpoint approximate area of 5.77476.

Section 5.1 – Estimating with Finite Sums

- Rectangular Approximation Method

15

5 sec

Lower Sum = Area of

inscribed = s(n)

Midpoint Sum

Upper Sum = Area

of circumscribed= S(n)

width of region

sigma = sum

y-value at xi

LRAM, MRAM, and RRAM approximations to the area under the graph of y=x2 from x=0 to x=3

Section 5.1 – Estimating with Finite Sums

- Rectangular Approximation Method (RAM)

(from Finney book)

y=x2

LRAM = Left-hand Rectangular

Approximation Method

= sum of (height)(width) of each

rectangle

height is measured on left side of

each rectangle

1 2 3

Section 5.1 – Estimating with Finite Sums

- Rectangular Approximation Method (cont.)

y=x2

RRAM = Right-hand Rectangular

Approximation Method

= sum of (height)(width) of each

rectangle

height is measured on right side

of rectangle

1 2 3

Section 5.1 – Estimating with Finite Sums

- Rectangular Approximation Method (cont.)

y=x2

MRAM = Midpoint Rectangular

Approximation Method

= sum of areas of each rectangle

height is determined by the height

at the midpoint of each horizontal region

1 2 3

Section 5.1 – Estimating with Finite Sums

- Estimating the Volume of a Sphere

The volume of a sphere can be estimated by a similar method using the sum of the volume of a finite number of circular cylinders.

definite_integrals.pdf (Slides 64, 65)

Section 5.1 – Estimating with Finite Sums

Cardiac Output problems involve the injection of dye into a vein, and monitoring the concentration of dye over time to measure a patient’s “cardiac output,” the number of liters of blood the heart pumps over a period of time.

Section 5.1 – Estimating with Finite Sums

See the graph below. Because the function is not known, this is an application of finite sums. When the function is known, we have a more accurate method for determining the area under the curve, or volume of a symmetric solid.

Section 5.1 – Estimating with Finite Sums

- Sigma Notation (from Larson book)

The sum of n terms

is written as

is the index of summation

is the ith term of the sum

and the upper and lower bounds of summation are n and 1 respectively.

Section 5.1 – Estimating with Finite Sums

- Properties of Summation

Section 5.1 – Estimating with Finite Sums

- Summation Formulas:

Section 5.1 – Estimating with Finite Sums

- Limit of the Lower and Upper Sum

If f is continuous and non-negative on the interval [a, b], the limits as of both the lower and upper sums exist and are equal to each other

Section 5.1 – Estimating with Finite Sums

- Definition of the Area of a Region in the Plane

Let f be continuous an non-negative on the interval [a, b]. The area of the region bounded by the graph of f, the x-axis, and the vertical lines x=a and x=b is

(ci, f(ci))

xi-1

xi

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