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

Disks 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 f(x)

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 = x 3 on the interval 0 < x < 2 rotated about x-axis

Revolve About Line Not a Coordinate Axis Consider the function y = 2x 2 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) a b g(x)

Application Given two functions y = x 2, and y = x 3  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 = x 2 and y = 4x, revolved about the y-axis  Radius of inner circle? f(y) = y/4  Radius of outer circle?  Limits? 0 < y < 16

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 c f(x) b a

Cross Sections This suggests a limit and an integral c f(x) b a

Cross Sections We could do similar summations (integrals) for other shapes  Triangles  Semi-circles  Trapezoids c f(x) b a

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 7.2A Page 463 Exercises 1 – 29 odd Lesson 7.2B Page 464 Exercises odd, 49, 53, 57