The Area Between Two Curves

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