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AP CALCULUS AB

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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  1. 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) = 412.5 miles t 2 5 8

  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 5.77476.

  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 1 2 3

  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 1 2 3

  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 1 2 3

  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

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