AP CALCULUS AB Chapter 5 The Definite Integral

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

Calculus Date: 2/10/2014 ID Check Objective: Estimating with Finite Sums Do Now: pg 274 QR 8 -10 HW Requests: Textbook QR 6. 1 #1 -10 Textbook pg 274 #1, 3, 5 -11 EX: pg 274 #13, 14, 15, 22 In Class: pg 274 #16, 20, 23 HW: Textbook pg 274 #16, 17 -25 odds 25 xtra credit -Bring your calculators for TI App Download Casio- please see if you have RAM program Announcements: Saturday Sessions Rm 241 10 -12: 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! Photo by Vickie Kelly, 2002 Greenfield Village, Michigan By Dr. Benjamin Elijah Mays, Greg Hanford High School, Richland, Washington Past. Kelly, 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: Life Is Just A Minute HW: Textbook QR 6. 1 #1 -10 Be careful of units and Life is just a minute—only sixty in it. Conversions Read Sectionseconds 6. 1 Forced upon you—can't refuse it. Bring your calculators. Didn't seek it—didn't choose it. Casio- please see if you have But it's up to you to use it. RAM program You must suffer if you lose it. Announcements: Give an account if you abuse it. Saturday Sessions Just a tiny, little minute,

Section 6. 1 – Estimating with Finite Sums p 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 6. 1 – Estimating with Finite Sums p 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 p 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.

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

Section 6. 1 – Estimating with Finite Sums p Rectangular Approximation Method 15 5 sec Lower Sum = Area of Upper Sum = Area inscribed = s(n) of circumscribed= S(n) width of region sigma = sum y-value at xi Midpoint Sum

What you’ll learn about Distance Traveled p Rectangular Approximation Method (RAM) p Volume of a Sphere p Cardiac Output p … and why Learning about estimating with finite sums sets the foundation for understanding integral calculus.

Section 5. 1 – Estimating with Finite Sums p 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) 75 mph TDT = Area under line = 3(75) = 225 miles 2 5 t

Section 5. 1 – Estimating with Finite Sums p Distance Traveled at Non-Constant Velocity: v(t) 75 Total Distance Traveled = Area of geometric figure = (1/2)h(b 1+b 2) = (1/2)75(3+8) = 412. 5 miles 2 5 8 t

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.

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 p 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 1 2 3

Section 5. 1 – Estimating with Finite Sums p 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 p Rectangular Approximation Method (cont. ) y=x 2 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 p 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 p Examples:

Section 5. 1 – Estimating with Finite Sums p 1. 2. Properties of Summation

Section 5. 1 – Estimating with Finite Sums p 1. 2. 3. 4. Summation Formulas:

Section 5. 1 – Estimating with Finite Sums p Example:

Section 5. 1 – Estimating with Finite Sums p 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 p 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