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

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002 Greenfield Village, Michigan EX: pg274 #13,14,15,22 In Class: pg 274 #16, 20, 23 HW: Textbook pg 274 #16, odds 25 xtra credit Life Is Just A Minute Life is just a minute—only sixty seconds in it. Forced upon you—can't refuse it. Didn't seek it—didn't choose it. But it's up to you to use it. You must suffer if you lose it. Give an account if you abuse it. Just a tiny, little minute, But eternity is in it! By Dr. Benjamin Elijah Mays, Past President of Morehouse College

Calculus Date: 2/7/2014 ID Check Objective: SWBAT complete assessment on Optimization and related rates problems. Do Now: Quiz Section 5.4 and 5.6 Turn in SM and Textbook Homework on these sections HW Requests: HW: Textbook QR 6.1 #1-10 Be careful of units and Conversions Read Section 6.1 Bring your calculators. Casio- please see if you have RAM program Announcements: Saturday Sessions Rm :50 Life Is Just A Minute Life is just a minute—only sixty seconds in it. Forced upon you—can't refuse it. Didn't seek it—didn't choose it. But it's up to you to use it. You must suffer if you lose it. Give an account if you abuse it. Just a tiny, little minute, But eternity is in it! By Dr. Benjamin Elijah Mays, Past President of Morehouse College

Section 6.1 – 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.pdfdefinite_integrals.pdf (Slides 64, 65)

Section 6.1 – 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.

Circumscribed rectangles are all above the curve: Inscribed rectangles are all below the curve:

Section 6.1 – Estimating with Finite Sums  Rectangular Approximation Method 15 5 sec Lower Sum = Area of inscribed = s(n) Upper Sum = Area of circumscribed= S(n) Midpoint Sum sigma = sumy-value at x i width of region

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? 25 75mph t v(t) TDT = Area under line = 3(75) = 225 miles

Section 5.1 – Estimating with Finite Sums  Distance Traveled at Non-Constant Velocity: t v(t) Total Distance Traveled = Area of geometric figure = (1/2)h(b 1 +b 2 ) = (1/2)75(3+8) = miles

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

LRAM, MRAM, and RRAM approximations to the area under the graph of y=x 2 from x=0 to x=3

Section 5.1 – Estimating with Finite Sums  Rectangular Approximation Method (RAM) (from Finney book) y=x 2 LRAM = Left-hand Rectangular Approximation Method = sum of (height)(width) of each rectangle height is measured on left side of each rectangle

Section 5.1 – Estimating with Finite Sums  Rectangular Approximation Method (cont.) y=x 2 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=x MRAM = Midpoint Rectangular Approximation Method = sum of areas of each rectangle height is determined by the height at the midpoint of each horizontal region

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 i th 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 1. 2.

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 (c i, f(c i )) x i-1 xixi