6.3 Volumes of Revolution Tues 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 #

6.3 Solids of Revolution Wed 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