1 / 92

The Area Between Two Curves

The Area Between Two Curves. Lesson 6.1. When f(x) < 0. Consider taking the definite integral for the function shown below. The integral gives a ___________ area We need to think of this in a different way. a. b. f(x). Another Problem.

leanna
Download Presentation

The Area Between Two Curves

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. The Area Between Two Curves Lesson 6.1

  2. When f(x) < 0 • Consider taking the definite integral for the function shown below. • The integral gives a ___________ area • We need to think of this in a different way a b f(x)

  3. Another Problem • What about the area between the curve and the x-axis for y = x3 • What do you get forthe integral? • Since this makes no sense – we need another way to look at it

  4. Solution • We can use one of the properties of integrals • We will integrate separately for _________ and __________ We take the absolute value for the interval which would give us a negative area.

  5. General Solution • When determining the area between a function and the x-axis • Graph the function first • Note the ___________of the function • Split the function into portions where f(x) > 0 and f(x) < 0 • Where f(x) < 0, take ______________ of the definite integral

  6. Try This! • Find the area between the function h(x)=x2 – x – 6 and the x-axis • Note that we are not given the limits of integration • We must determine ________to find limits • Also must take absolutevalue of the integral sincespecified interval has f(x) < 0

  7. Area Between Two Curves • Consider the region betweenf(x) = x2 – 4 and g(x) = 8 – 2x2 • Must graph to determine limits • Now consider function insideintegral • Height of a slice is _____________ • So the integral is

  8. The Area of a Shark Fin • Consider the region enclosed by • Again, we must split the region into two parts • _________________ and ______________

  9. Slicing the Shark the Other Way • We could make these graphs as ________________ • Now each slice is_______ by (k(y) – j(y))

  10. Practice • Determine the region bounded between the given curves • Find the area of the region

  11. Horizontal Slices • Given these two equations, determine the area of the region bounded by the two curves • Note they are x in terms of y

  12. Assignments A • Lesson 7.1A • Page 452 • Exercises 1 – 45 EOO

  13. Integration as an Accumulation Process • Consider the area under the curve y = sin x • Think of integrating as an accumulation of the areas of the rectangles from 0 to b b

  14. Integration as an Accumulation Process • We can think of this as a function of b • This gives us the accumulated area under the curve on the interval [0, b]

  15. Try It Out • Find the accumulation function for • Evaluate • F(0) • F(4) • F(6)

  16. Applications • The surface of a machine part is the region between the graphs of y1 = |x| and y2 = 0.08x2 +k • Determine the value for k if the two functions are tangent to one another • Find the area of the surface of the machine part

  17. Assignments B • Lesson 7.1B • Page 453 • Exercises 57 – 65 odd, 85, 88

  18. Volumes – The Disk Method Lesson 7.2

  19. 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

  20. 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?

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

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

  23. 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 radius_______________

  24. 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

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

  26. Revolving About y-Axis • Also possible to revolve a function about the y-axis • Make a disk or a washer to be ______________ • Consider revolving a parabola about the y-axis • How to represent the radius? • What is the thicknessof the disk?

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

  28. 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) = _____ • Radius of outer circle? • Limits? • 0 < y < 16

  29. 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

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

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

  32. 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

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

  34. Volume: The Shell Method Lesson 7.3

  35. 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.

  36. If we take a ____________slice and revolve it about the y-axis we get a cylinder.

  37. Shell Method • Based on finding volume of cylindrical shells • Add these volumes to get the total volume • Dimensions of the shell • _________of the shell • _________of the shell • ________________

  38. 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)

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

  40. Hints for Shell Method • Sketch the __________over the limits of integration • Draw a typical __________parallel to the axis of revolution • Determine radius, height, thickness of shell • Volume of typical shell • Use integration formula

  41. 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 = _____ _______________ = y

  42. 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

  43. 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

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

  45. Try It Out • Integral for the volume is

  46. Assignment • Lesson 7.3 • Page 472 • Exercises 1 – 25 odd • Lesson 7.3B • Page 472 • Exercises 27, 29, 35, 37, 41, 43, 55

  47. Arc Length and Surfaces of Revolution Lesson 7.4

  48. Why? Arc Length • We seek the distance along the curve fromf(a) to f(b) • That is from P0 to Pn • The distance formula for each pair of points P1 Pi Pn • P0 • • • • • b a What is another way of representing this?

  49. Arc Length • We sum the individual lengths • When we take a limit of the above, we get the integral

  50. Arc Length • Find the length of the arc of the function for 1 < x < 2

More Related