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Review Problem – Riemann Sums

Review Problem – Riemann Sums. Use a right Riemann Sum with 3 subintervals to approximate the definite integral:. Applications of the Definite Integral. Mr. Reed AP Calculus AB. Finding Areas Bounded by Curves. To get the physical area bounded by 2 curves:

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Review Problem – Riemann Sums

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  1. Review Problem – Riemann Sums • Use a right Riemann Sum with 3 subintervals to approximate the definite integral:

  2. Applications of the Definite Integral Mr. Reed AP Calculus AB

  3. Finding Areas Bounded by Curves • To get the physical area bounded by 2 curves: • Graph curves & find intersection points – limits of integration • Identify “top” curve & “bottom” curve OR “right-most” curve & “left-most” curve • Draw a representative rectangle • Set up integrand: • Top – Bottom • Right – Left

  4. Finding Intersection Points • Set equations equal to each other and solve algebraically • Graph both equations and numerically find intersection points

  5. Example #1 Find the area of the region between y = sec2x and y = sinx from x = 0 to x = pi/4.

  6. Example #2 Find the area that is bounded between the horizontal line y = 1 and the curve y = cos2x between x = 0 and x = pi.

  7. Example #3 • From Text – p.240 - #16

  8. Example #4 Find the area of the region R in the first quadrant that is bounded above by y = sqrt(x) and below by the x-axis and the line y = x – 2.

  9. Summarize the process

  10. AP MC Area Problem • #12 from College Board Course Description

  11. Homework • P.236-240: Q1-Q10, 13-25(odd)

  12. Authentic Applications for the Definite Integral • Example  #2 – p.237

  13. Definite Integral Applied to Volume • 2 general types of problems: • Volume by revolution • Volumes by base

  14. Volume by Revolution – Disk Method The region under the graph of y = sqrt(x) from x = 0 to x = 2 is rotated about the x-axis to form a solid. Find its volume.

  15. Volume by Revolution – Disk Method

  16. Homework #1 – Disk Method about x and y axis • P.246-247: Q1-Q10,1,3,5

  17. Volume by Revolution – About another axis The region bounded by y = 2 – x^2 and y = 1 is rotated about the line y = 1. Find the volume of the resulting solid.

  18. Volume by Revolution – Washer Method Find the volume of the solid formed by revolving the region bounded by the graphs of f(x) = sqrt(x) and g(x) = 0.5x about the x-axis.

  19. Homework #2 – Washer Method & Different axis • P.247 – 249: 7,9,11,14

  20. Volume with known base The base of a solid is given by x^2 + y^2 = 4. Each slice of the solid perpendicular to the x-axis is a square. Find the volume of the solid.

  21. Homework #3 – Different axis & known base • P.249: 15,16,18,19

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