6 3 volumes of revolution mon march 10
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6.3 Volumes of Revolution Mon March 10. Do Now The volume of the solid whose base is the region enclosed by y = x^2 and y = 3, and whose cross sections perpendicular to the y-axis are rectangles of height y^3. Solid of revolution.

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6 3 volumes of revolution mon march 10
6.3 Volumes of RevolutionMon March 10
  • Do Now
  • The volume of the solid whose base is the region enclosed by y = x^2 and y = 3, and whose cross sections perpendicular to the y-axis are rectangles of height y^3
solid of revolution
Solid of revolution
  • A solid of revolution is a solid obtained by rotating a region in the plan about an axis
  • Pic:
  • The cross section of these solids are circles
disk method
Disk Method
  • If f(x) is continuous and f(x) >= 0 on [a,b] then the solid obtained by rotating the region under the graph about the x-axis has volume
slide4
Ex
  • Calculate the volume V of the solid obtained by rotating the region under y = x^2 about the x-axis for [0,2]
washer method
Washer Method
  • If the region rotated is between 2 curves, where f(x) >= g(x) >= 0, then
slide6
Ex
  • Find the volume V obtained by revolving the region between y = x^2 + 4 and y = 2 about the x-axis for [1,3]
revolving about any horizontal line
Revolving about any horizontal line
  • When revolving about a horizontal line that isn’t y = 0, you have to consider the distance from the curve to the line.
  • Ex: if you were revolving y = x^2 about y = -1, then the radius would be (x^2 + 1)
slide8
Ex
  • Find the volume V of the solid obtained by rotating the region between the graphs of

f(x) = x^2 + 2 and g(x) = 4 – x^2 about the line y = -3

revolving about a vertical line
Revolving about a vertical line
  • If you revolve about a vertical line, everything needs to be in terms of y!
    • Y – bounds
    • Curves in terms of x = f(y)
    • There is no choice between x or y when it comes to volume!
slide10
Ex
  • Find the volume of the solid obtained by rotating the region under the graph of

f(x) = 9 – x^2 for [0,3] about the line x = -2

closure
Closure
  • Find the volume obtained by rotating the graphs of f(x) = 9 – x^2 and y = 12 for [0,3] about the line y = 15
  • HW: p.381 #1-53 EOO
6 3 solids of revolution tues march 11
6.3 Solids of RevolutionTues March 11
  • Do Now
  • Find the volume of the solid obtained by rotating the region between y = 1/x^2 and the x – axis over [1,4] about the x-axis
solids of revolution
Solids of Revolution
  • Disk Method: no gaps
  • Washer Method: gaps
    • Outer – Inner
    • Radii depend on the axis of revolution
    • In terms of x or y depends on horizontal or vertical lines of revolution
closure1
Closure
  • Find the volume of the solid obtained by rotating the region enclosed by y = 32 – 2x, y = 2 + 4x, and x = 0, about the y - axis
  • HW: p.381 #1-53 AOO
  • 6.1-6.3 Quiz on Friday (I won’t be in class Thurs, but will be available at break, 8th period, or after school
6 3 solids of revolution review wed march 12
6.3 Solids of Revolution ReviewWed March 12
  • Do Now
  • Find the volume of the solid obtained by rotating the region between y = x^2 and y = 2x + 3 about the x-axis
6 1 6 3 review questions graphing calculator set up integral
6.1-6.3 Review _ questionsGraphing Calculator = Set up integral
  • 6.1 Area between curves
    • In terms of x or y
    • Bounds - intersections
  • 6.2 Volume using cross sections / Average Value
    • V = Integral of area of cross sections
    • AV = Integral divided by length of interval
  • 6.3 Solids of Revolution
    • With respect to different lines
    • Disks vs Washers
closure2
Closure
  • HW: Ch 6 AP Questions MC #1-6 8-14 17 18 20 FRQ #1 2
  • Answers on powerpoint
  • 6.1-6.3 Quiz Fri
6 1 6 3 review
6.1-6.3 Review
  • AP Answers (even):
  • 2) D 14) E
  • 4) C 18) A
  • 6) C 20) B
  • 8) D 2a)
  • 10) C b)
  • 12) C c)
6 1 6 3 review1
6.1-6.3 Review
  • Ch 6 AP Worksheet
  • 1) D 6c) 5.470
  • 2) C 6d) 0.029
  • 3) E 7a) 1382.954 hours
  • 4) B 7b) increasing s’(100) = .029
  • 5) D 7c) 13.094 hours/day
  • 6a) .307 7d) 165th day
  • 6b) 1.119
6 1 6 3 review 6 questions graphing calculator set up integral
6.1-6.3 Review 6 questionsGraphing Calculator = Set up integral
  • 6.1 Area between curves
    • In terms of x or y
    • Bounds - intersections
  • 6.2 Volume using cross sections / Average Value
    • V = Integral of area of cross sections
    • AV = Integral divided by length of interval
  • 6.3 Solids of Revolution
    • With respect to different lines
    • Disks vs Washers
closure3
Closure
  • Which application of the integral do you imagine would be the most useful in real world applications? Why?
  • 6.1-6.3 Quiz tomorrow!
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