Integration TERMINOLOGY. Definite integral: The integral or primitive function restricted to a lower

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1 TERMINOLOGY Integration Definite integral: The integral or primitive function restricted to a lower b and upper boundary. It has the notation f (x).dx and geometrically a represents the area between the curve y = f (x), the x axis and the ordinates x = a and x = b Even function: A function where f ( x) = f (x). It is symmetrical about the y axis Indefinite integral: General primitive function represented by f (x). dx Integration: The process of finding a primitive function Odd function: A function where f ( x) = f (x). An odd function has rotational symmetry about the origin watch a movie time for Calculus Board of Studies 1

2 What is Integration? Integration Integration is the process of finding an area under a curve. It is an important process in many areas of knowledge, such as surveying, physics and the social sciences. We will be looking at approximation methods of integrating, as well as shorter methods that lead to finding areas and volumes. Archimedes ( BC) found the area of enclosed curves by cutting them into very thin layers and finding their sum. He found the formula for the volume of a sphere this way. He also found an estimation of π, correct to 2 decimal places. 2

3 Approximate Methods Integration To find an area under a curve, we can use rectangles in order to find the approximate area between a curve and the x axis. The area of each rectangle is f (x) δ x where f (x) is the height and δx is the width of each rectangle. As δx 0, sum of rectangles exact area. Area = lim Σ f (x). δx δx 0 = f(x). dx Now there are other, more accurate ways to find the area under a curve. However, the notation is still used. geogebra activity 3

4 Trapezoidal rule Integration The trapezoidal rule uses a trapezium for the approximate area under a curve. A trapezium generally gives a better approximation to the area than a rectangle, and can sometimes be used if a definite integral (more on that later) is too difficult to calculate. 4

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6 Integration There is a more general formula for n subintervals. Several trapezia give a more accurate area than one. However, you could use the first formula several times if you prefer using it. 6

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10 Simpson's Rule Integration This is generally more accurate than the trapezoidal rule, since it makes use of parabolic arcs instead of straight lines. A parabola is drawn through points A, B and C to give the formula: One application of Simpson s rule uses 3 function values (ordinates). Two applications use 5 function values, three applications uses 7 function values. 10

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12 Integration There is another Simpson's Rule, used for multiple applications, although I feel it is just easier to use the basic rule with a number of sections... You could also remember this version as: f(x). dx = h [(y0 + yn) = 4(odds) + 2 (evens)] 12

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14 Fundamental Theorem of Calculus Integration Mathematicians found a link between finding areas under a curve and the primitive function. This made possible a simple method for finding exact areas. 14

15 Proof: Integration Consider a continuous curve y = f (x) for all values of x > a Let area ABCD be A(x) Let area ABGE be A(x+h) Then area DCGE is A(x+h) A(x) area DCFE < area DCGE<area DHGE i.e. f(x).h < A(x+h) A(x)< f(x+h).h f(x) < A(x+h) A(x)< f(x+h) h lim f(x) < lim A(x+h) A(x) < lim f(x+h) h 0 h 0 h f(x) < A' (x) < f(x) So A'(x) = f(x) h 0 A(x) is a primitive function of f(x) A(x) = F(x) + C where F(x) is the primitive function of f(x) (1) Now, A(x) is the area between a and x A(a) = 0 Substitute in (1) A(a) = F(a) + C 0 = F(a) + C F(a) = C A(x) = F(x) F(a) If x=b where b>a, A(b) = F(b) F(a) 15

16 By the fundamental theorem of calculus, where F(x) is the primitive function of f(x). The primitive function of x is Integration Putting these pieces of information together, we can find areas under simple curves. 16

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22 Indefinite Integrals Sometimes it is necessary to find a general or indefinite integral (primitive function). Integration 22

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30 Areas Enclosed by the x-axis Integration The definite integral gives the signed area under a curve. Areas above the x axis give a positive definite integral. Areas below the x axis give a negative definite integral. We normally think of areas as positive. So to find areas below the x axis, take the absolute value of the definite integral. That is, geogebra activity 30

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33 Odd and Even functions Integration 33

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37 Areas Enclosed by the y-axis Integration To find the area between a curve and the y axis, we change the subject of the equation of the curve to x. That is, x = f (y). The definite integral is given by Since x is positive on the right hand side of the y axis, the definite integral is positive. Since x is negative on the left hand side of the y axis, the definite integral is negative. 37

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41 Sums and differences of Areas 41

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44 Volumes of Solids of Revolution Integration The volume of a solid can be found by rotating an area under a curve about the x axis or the y axis. We generally refer to this as 'the volume of a solid of revolution'. watch a movie volume geogebra activity curve visualiser 44

45 Volumes about the x-axis Integration Pull To see WHY this is true, watch the video on the previous page. It has to do with finding the volume of a very thin cylinder and then taking an infinite number of successively thinner cylinders to approximate the area. More cylinders?better approximation. For the proof see the sidebar. 45

46 Volumes around the y-axis Integration The formula for rotations about the y axis is similar to the formula for the x axis. 46

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51 Integration using Substitution Integration EXTENSION CONTENT!!! Sometimes harder integrals can be found by using a substitution method. By choosing an appropriate variable, we can simplify the function to be integrated. 51

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59 Notes: Integration Be able to find definite and indefinite integrals and understand the relationship between integration and area Be able to find the volume of a solid rotated around the x or the y axis SPECIAL NOTE: Part of this topic is integrating trigonometric functions...but that will have to wait until a bit later! Make your summary, find and attempt some HSC level questions 59

60 Attachments jw_upper_and_lower_sums_using_rectangles.ggb