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AP CALCULUS AB. Chapter 5: The Definite Integral Section 5.1: Estimating with Finite Sums. What you’ll learn about. Distance Traveled Rectangular Approximation Method (RAM) Volume of a Sphere Cardiac Output … and why

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Ap calculus ab

AP CALCULUS AB

Chapter 5:

The Definite Integral

Section 5.1:

Estimating with Finite Sums


What you ll learn about
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
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 sums1
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 finding distance traveled when velocity varies
Example Finding Distance Traveled when Velocity Varies


Example finding distance traveled when velocity varies1
Example Finding Distance Traveled when Velocity Varies


Example estimating area under the graph of a nonnegative function
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 sums2
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



Section 5 1 estimating with finite sums3
Section 5.1 – Estimating with Finite Sums graph of

  • 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 sums4
Section 5.1 – Estimating with Finite Sums graph of

  • 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 sums5
Section 5.1 – Estimating with Finite Sums graph of

  • 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 sums6
Section 5.1 – Estimating with Finite Sums graph of

  • 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 sums7
Section 5.1 – Estimating with Finite Sums graph of

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 sums8
Section 5.1 – Estimating with Finite Sums graph of

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 sums9
Section 5.1 – Estimating with Finite Sums graph of

  • 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 sums11
Section 5.1 – Estimating with Finite Sums graph of

  • Properties of Summation


Section 5 1 estimating with finite sums12
Section 5.1 – Estimating with Finite Sums graph of

  • Summation Formulas:



Section 5 1 estimating with finite sums14
Section 5.1 – Estimating with Finite Sums graph of

  • 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 sums15
Section 5.1 – Estimating with Finite Sums graph of

  • 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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