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Volumes

Volumes. Lesson 6.2. 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 Δ x What is the volume of the slab? Now sum the volumes of all such slabs. f(x). c. a. b. Cross Sections. This suggests a limit and an integral.

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Volumes

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  1. Volumes Lesson 6.2

  2. 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

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

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

  5. 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

  6. 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

  7. 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?

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

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

  10. 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

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

  12. 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?

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

  14. 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

  15. Assignment • Lesson 6.2A • Page 372 • Exercises 1 – 29 odd

  16. 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 slice directly.

  17. If we take a vertical slice and revolve it about the y-axis we get a cylinder.

  18. Shell Method • Based on finding volume of cylindrical shells • Add these volumes to get the total volume • Dimensions of the shell • Radius of the shell • Thickness of the shell • Height

  19. The Shell • Consider the shell as one of many of a solid of revolution • The volume of the solid made of the sum of the shells dx f(x) f(x) – g(x) x g(x)

  20. Try It Out! • Consider the region bounded by x = 0, y = 0, and

  21. Hints for Shell Method • Sketch the graph over the limits of integration • Draw a typical shell parallel to the axis of revolution • Determine radius, height, thickness of shell • Volume of typical shell • Use integration formula

  22. Rotation About x-Axis • Rotate the region bounded by y = 4x and y = x2 about the x-axis • What are the dimensions needed? • radius • height • thickness thickness = dy radius = y

  23. Rotation About Non-coordinate Axis • Possible to rotate a region around any line • Rely on the basic concept behind the shell method g(x) f(x) x = a

  24. Rotation About Non-coordinate Axis • What is the radius? • What is the height? • What are the limits? • The integral: r g(x) f(x) a – x x = c x = a f(x) – g(x) c < x < a

  25. r = 2 - x 4 – x2 Try It Out • Rotate the region bounded by 4 – x2 , x = 0 and, y = 0 about the line x = 2 • Determine radius, height, limits

  26. Try It Out • Integral for the volume is

  27. Assignment • Lesson 6.2B • Page 373 • Exercises 41 – 59 odd

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