AP Calculus BC April 3-4, 2017 Mrs. Agnew Volumes of Revolution AP Calculus BC April 3-4, 2017 Mrs. Agnew

Essential Question Essential Vocabulary How do find the volumes of solids using the washer method and the disk method? Essential Vocabulary Volumes of Revolution Washer Method Disk Method

Find the Given Volumes Find the volume of the solid formed when f(x) is rotated about the given axis y = 3 x axis between x = -2 and x = 3 y = 4x y axis between x = 0 and x = 4 y = √25 – x2 x axis The above problems turn out to be easy since the solid becomes a standard geometric figure.

Volumes of Solids What is the general method for finding the volume of any solid? Assume we are given a solid that is not a familiar shape from Geometry. Start by finding a “cross section” of the solid Find the area of the cross section “Cut” the solid into n cross sectional areas Find the sum of volumes of each piece

Volumes of Solids So pretty easy process, right?.... If A(x) is the area of the cross section, the volume would be the area of the cross section times the height of each “piece.” So the volume then is…

Volumes of Solids Look familiar? Just as we did for area, we can rewrite the previous formula using integrals… When finding the area of a region bound by the curve and x axis, we used rectangles. What should we use for volume???

Volumes of Solids Just as we used rectangles to find the area of a region, we will use cylinders (rectangles with a height) to find volumes of solids. Guided Practice: page 463 #2, 4

Disks vs. Washers The disk method can be used when the cross section is a disk… A = πr2 If we want to revolve a region bounded by two curves, we will need to use the washer method. Find the inner radius & outer radius, and then find the area… A = π(outer radius)2 – π(inner radius)2

Practice and Homework Guided Practice Homework page 463 #6 – 16 (Even) page 463 #1 – 15 (Odd)