Ap calculus ab
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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

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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

    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 Finding Distance Traveled when Velocity Varies


Example Finding Distance Traveled when Velocity Varies


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

  • Examples:


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

  • Example:


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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