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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

- 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

- 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

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

- If the region rotated is between 2 curves, where f(x) >= g(x) >= 0, then

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

- 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)

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

- 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!

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

- 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 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

- 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

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 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 _ 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

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

- 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 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 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

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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