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 2 5 t 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 2 5 8 t Example Finding Distance Traveled when Velocity Varies A p article starts at x 0 an d m o ves alo n g th e x -ax is w ith velo city v ( t ) t 2 fo r tim e t 0 . W h ere is th e p article at t 3? G raph v and partition the tim e interval into subintervals of length t . If you us e t 1 / 4, you w ill have 12 subintervals. T he area of each rectangle approxim ates the distance traveled over the subint erval. A dding all of the areas (distanc es) gives an approxim ation to the total area under the curve (total distance travele d) from t 0 to t 3. Example Finding Distance Traveled when Velocity Varies C ontinuing in this m anner, derive the ar ea 1 / 4 m for each subinterval and 2 i add them : 1 256 8.98 9 256 25 256 49 256 81 256 121 256 169 256 225 256 289 256 361 256 441 256 529 256 2300 256 Example Estimating Area Under the Graph of a Nonnegative Function E stim ate the area under the graph of f ( x ) x s in x from x 0 to x 3. 2 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 Upper Sum = Area inscribed = s(n) of circumscribed= S(n) s n Area of region S n n A f x x i i 1 sigma = sum Midpoint Sum width of region 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 2 2 2 1 1 1 3 1 5 1 2 1 2 1 LRAM 0 1 2 2 2 2 2 2 2 2 2 2 2 6 . 875 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 RRAM 1 2 23 2 2 1 1 3 1 5 1 2 1 2 1 2 1 1 2 3 2 2 2 2 2 2 2 2 2 11 . 375 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 2 MRAM 2 2 2 2 2 1 1 3 1 5 1 7 1 9 1 11 1 4 2 4 2 4 2 4 2 4 2 4 2 8 . 9375 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 a1 , a 2 , a 3 ,..., a n is written as n a i a 1 a 2 a 3 ... a n i 1 is the index of summation a i is the ith term of the sum and the upper and lower bounds of summation are n and 1 respectively. i Section 5.1 – Estimating with Finite Sums Examples: 5 i 1 2 3 4 5 i 1 n i i 1 2 1 1 1 2 1 3 1 ... n 1 2 2 2 2 Section 5.1 – Estimating with Finite Sums Properties of Summation n 1. ka i 1 n 2. a i 1 i n i k ai i 1 bi n n a b i i 1 i 1 i Section 5.1 – Estimating with Finite Sums Summation Formulas: n 1. 2. c cn i 1 n i i 1 3. 4. n i 2 i 1 n i i 1 3 n n 1 2 n n 1 2 n 1 6 2 2 n n 1 4 Section 5.1 – Estimating with Finite Sums Example: 10 10 i i i 1 2 i 1 i 3 i i 1 10 10 1 2 4 10 i 1 i i 1 10 3 100 121 4 2 10 10 1 10 11 2 25 121 5 11 3080 2 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 n of both the lower and upper sums exist and are equal to each other lim s n lim n n where x n f m x lim f M x lim S n i i 1 ba n and maximum n and n i i 1 n f m i and f M i are the minimum values of f on the i th subinterva l. 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 n Area lim n and x f c x , i i 1 x i 1 c i x i (ci, f(ci)) ba n xi-1 xi