1. 6.3 Volumes of Revolution Fri Feb 26 Do Now Find 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

2. Solid of revolution A solid of revolution is a solid obtained by rotating a region in the plane about an axis Pic: The cross section of these solids are circles

3. 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

4. Ex Calculate the volume V of the solid obtained by rotating the region under y = x^2 about the x-axis for [0,2]

5. Washer Method If the region rotated is between 2 curves, where f(x) >= g(x) >= 0, then

6. 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]

7. 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)

8. 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

9. 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!

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

11. 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

12. 6.3 Solids of Revolution Mon Feb 29 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

13. HW Review: p.381 #1-53

14. 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

15. 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 Thurs?

16. 6.3 Solids of Revolution Review Tues March 1 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

17. HW Review: p.381 #1-53

18. 6.1-6.3 Review _ questions Graphing 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

19. 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 Thurs

20. 6.1-6.3 Review Wed March 2 Do Now 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

21. 6.1-6.3 Review AP Answers (even): 2)D14) E 4)C18) A 6)C20) B 8)D2a) 10) Cb) 12) Cc)

22. 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) 165 th day 6b) 1.119

23. 6.1-6.3 Review 6 questions Graphing 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

24. 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