1. Estimating with Finite Sums
AP CALCULUS AB
Chapter 5:
The Definite Integral
Section 5.1:
Estimating with Finite Sums

2. 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.

3. 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

4. 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)
= miles t

5. Example Finding Distance Traveled when Velocity Varies

6. Example Finding Distance Traveled when Velocity Varies

7. 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

8. 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

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

10. 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

11. 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

12. 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

13. 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)

14. 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.

15. 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.

16. 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.

17. Section 5.1 – Estimating with Finite Sums
Examples:

18. Section 5.1 – Estimating with Finite Sums
Properties of Summation

19. Section 5.1 – Estimating with Finite Sums
Summation Formulas:

20. Section 5.1 – Estimating with Finite Sums
Example:

21. 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

22. 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