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Volumes – The Disk Method

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. dx. Disks. f(x).

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Volumes – The Disk Method

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  1. Volumes – The Disk Method Lesson 7.2

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

  3. dx Disks f(x) • We seek ways of usingintegrals to determine thevolume 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?

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

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

  6. 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 forthe radiusin terms of y

  7. 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

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

  9. 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 thicknessof the disk?

  10. Revolving About y-Axis • Must consider curve asx = f(y) • Radius = f(y) • Slice is dy thick • Volume of the solid rotatedabout y-axis

  11. 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

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

  13. Cross Sections • This suggests a limitand an integral f(x) c a b

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

  15. 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

  16. 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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