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Volumes – The Disk Method Lesson 7.2

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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

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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)

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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

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Try It Out! Try the function y = x 3 on the interval 0 < x < 2 rotated about x-axis

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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

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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)

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Application Given two functions y = x 2, and y = x 3 Revolve region between about x-axis What will be the limits of integration?

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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?

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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

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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

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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

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

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

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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

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Assignment Lesson 7.2A Page 463 Exercises 1 – 29 odd Lesson 7.2B Page 464 Exercises 31 - 39 odd, 49, 53, 57

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Volume: The Shell Method Lesson 7.3. Find the volume generated when this shape is revolved about the y axis. We can’t solve for x, so we can’t use a horizontal.

Volume: The Shell Method Lesson 7.3. Find the volume generated when this shape is revolved about the y axis. We can’t solve for x, so we can’t use a horizontal.

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