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

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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


6 3 volumes of revolution mon march 10

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


6 3 volumes of revolution mon march 10

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)


6 3 volumes of revolution mon march 10

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!


6 3 volumes of revolution mon march 10

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


Hw review p 381 1 53

HW Review: p.381 #1-53


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


Hw review p 381 1 531

HW Review: p.381 #1-53


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)D14) E

  • 4)C18) A

  • 6)C20) B

  • 8)D2a)

  • 10) Cb)

  • 12) Cc)


6 1 6 3 review1

6.1-6.3 Review

  • Ch 6 AP Worksheet

  • 1) D6c) 5.470

  • 2) C6d) 0.029

  • 3) E7a) 1382.954 hours

  • 4) B7b) increasing s’(100) = .029

  • 5) D7c) 13.094 hours/day

  • 6a) .3077d) 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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