Volumes – The Disk Method Lesson 7.2

Revolving a Function Consider a function f(x) on the interval [a, b] Now consider revolving that segment of curve about the x axis What kind of functions generated these solids of revolution? f(x) a b

Revolving a Function What kind of function here?

Disks f(x) We seek ways of using integrals to determine the volume of these solids Consider a disk which is a slice of the solid What is the radius What is the thickness What then, is its volume? dx

Disks To find the volume of the whole solid we sum the volumes of the disks Shown as a definite integral f(x) a b

Try It Out! Try the function y = x3 on the interval 0 < x < 2 rotated about x-axis

Revolve About Line Not a Coordinate Axis Consider the function y = 2x2 and the boundary lines y = 0, x = 2 Revolve this region about the line x = 2 We need an expression for the radius in terms of y

Washers Consider the area between two functions rotated about the axis Now we have a hollow solid We will sum the volumes of washers As an integral f(x) g(x) a b

What will be the limits of integration? Application Given two functions y = x2, and y = x3 Revolve region between about x-axis What will be the limits of integration?

Revolving About y-Axis Also possible to revolve a function about the y-axis Make a disk or a washer to be horizontal Consider revolving a parabola about the y-axis How to represent the radius? What is the thickness of the disk?

Revolving About y-Axis Must consider curve as x = f(y) Radius = f(y) Slice is dy thick Volume of the solid rotated about y-axis

Flat Washer Determine the volume of the solid generated by the region between y = x2 and y = 4x, revolved about the y-axis Radius of inner circle? f(y) = y/4 Radius of outer circle? Limits? 0 < y < 16

Assignment A Lesson 7.2A Page 463 Exercises 1 – 29 odd Spreadsheet Assignment Due in 1 Week

Cross Sections Consider a square at x = c with side equal to side s = f(c) Now let this be a thin slab with thickness Δx What is the volume of the slab? Now sum the volumes of all such slabs

Cross Sections This suggests a limit and an integral

Cross Sections We could do similar summations (integrals) for other shapes Triangles Semi-circles Trapezoids

Try It Out Consider the region bounded above by y = cos x below by y = sin x on the left by the y-axis Now let there be slices of equilateral triangles erected on each cross section perpendicular to the x-axis Find the volume

Assignment Lesson 7.2B Page 464 Exercises 31 - 39 odd, 49, 53, 57