6.3 Volumes of RevolutionTues Dec 15

• 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

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

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 #7 14 23 25 27-32 41 47 53

6.3 Solids of RevolutionWed Dec 16

• 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

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

Practice AP FRQs

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: AP FRQs• Ch 6 Test Tues Dec 22