Understanding Definite Integration in Mathematics
Delve into the concept of definite integration in mathematics with a focus on finding areas under curves, evaluating definite integrals, and solving exercises that involve calculating areas between curves and the x-axis. Explore the principles and techniques behind finding total areas between specified limits using integration methods.
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C2 Chapter 11: Integration Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com Last modified: 1st September 2015
Recap 2?2+ 3? ?? =2 3?3+3 2?2+ ? ? 1 + ? + ?2+ ?3 ?? = ? +1 2?2+1 3?3+1 ? 4?4+ ? 3 ?? = 3 ? 7 4? 4 ? 3+ ? 2 + ? ?2 ?? = 2? 1 2? 1 2+ ? ? ? ? ?4 ?? = 2 3? 3 ? 2+ ?
Definite Integration ? Suppose you wanted to find the area under the curve between ? = ? and ? = ?. ? ?? ? ? We could add together the area of individual strips, which we want to make as thin as possible
Definite Integration ? ? = ?(?) ?? ? ?1 ? ?2 ?3 ?4 ?5 ?6 ?7 ? What is the total area between ?1 and ?7? ? 7 ? ? ?? ? ?? ?? ? As ?? 0 ?=1
Definite Integration ? ? ? ?? ? You could think of this as Sum the values of ?(?) between ? = ? and ? = ?. ? Reflecting on above, do you think the following definite integrals would be positive or negative or 0? ? = sin? ? 2sin ? ?? + 0 0 ? ? 2? 2? 0 + sin ? ?? 0 2? + 0 ? sin ? ?? 2
Evaluating Definite Integrals 2 2 = ?3 ? 3?2 ?? We use square brackets to say that we ve integrated the function, but we re yet to involve the limits 1 and 2. 1 1 = 23 13 = 7 ? ? Then we find the difference when we sub in our limits. ? ? ? ?? = ? ? ?= ? ? ?(?) ? ?
Evaluating Definite Integrals 1 2 4?3+ 3?2 ?? 2?3+ 2? ?? 2 = ?4+ ?3 = 1 1 16 8 = 8 1 1 2 1 2?4+ ?2 2 ? = 1 1 2+ 1 = 8 + 4 ? =21 Bro Tip: Be careful with your negatives, and use bracketing to avoid errors. 2
Exercise 11B Find the area between the curve with equation ? = ? ? the ?-axis and the lines ? = ? and ? = ?. 1 ? ? ? ? ? ? = 3?2 2? + 2 ? ? = ? + 2?? = 1,? = 2 ? ? = ?3+ ?? = 1,? = 4 ? = 0,? = 2 a c e ?.?? ??? ?? 8 The sketch shows the curve with equation y = ?(?2 4). Find the area of the shaded region (hint: first find the roots). 2 ? ? ??? ? Find the area of the finite region between the curve with equation ? = (3 ?)(1 + ?) and the ?-axis. 4 ? Find the area of the finite region between the curve with equation ? = ?22 ? and the ?-axis. 6 ?? ? ?
Harder Examples Find the area bounded between the curve with equation ? = ?3 ? and the ?-axis. ? Sketch: (Hint: factorise!) ? ? 1 1 1?3 ? ?? and why? Looking at the sketch, what is 1 0, because the positive and negative region cancel each other out. ? What therefore should we do? Find the negative and positive region separately. 1 So total area is ? ? 0 1?3 3 ?? = +1 ? ?3 3 ?? = 1 4 0 ?+? 4 ?=?
Harder Examples Sketch the curve with equation ? = ? ? 1 ? + 3 and find the area between the curve and the ?-axis. The Sketch The number crunching ? ? 1 ? + 3 = ?3 2?2 3? 0 ?3 2?2 3? ?? = 11.25 ? 3 1 ?3 2?2 3? ?? = 7 ? ? 12 0 ? 1 -3 7 12= 115 Adding: 11.25 + 6
Exercise 11C Find the area of the finite region or regions bounded by the curves and the ?-axis. 11 ? ? = ? ? + 2 1 3 205 ? ? = ? + 1 ? 4 2 6 401 ? ? = ? + 3 ? ? 3 3 2 11 ? ? = ?2? 2 4 3 211 ? ? = ? ? 2 ? 5 5 12
Curves bound between two lines ? = ?(?) ?? ? ? ? ??(?) meant the sum of all the ? values Remember that ? between ? = ? and ? = ? (by using infinitely thin strips).
Curves bound between two lines ? ? ? How could we use a similar principle if we were looking for the area bound between two lines? ? What is the height of each of these strips? ? ? ?(?) ? ? ? ? ? ? = ? therefore area ?
Curves bound between two lines Find the area bound between ? = ? and ? = ? 4 ? . ? 3 ? 4 ? ? ?? = 4.5 ? 0 ? Bro Tip: Always do the function of the top line minus the function of the bottom line. That way the difference in the ? values is always positive, and you don t have to worry about negative areas. Bro Tip: We ll need to find the points at which they intersect.
Curves bound between two lines Edexcel C2 May 2013 (Retracted) ? = 4, ? = 2 ? ? Area = 36
More complex areas Bro Tip: Sometimes we can subtract areas from others. e.g. Here we could start with the area of the triangle OBC. C A B ???? = ??? ? ?
Exercise 11D ? 2,6 ? 2,6 ???? = 102 ? A region is bounded by the line ? = 6 and the curve ? = ?2+ 2. a) Find the coordinates of the points of intersection. b) Hence find the area of the finite region bounded by ?? and the curve. 1 3 The diagram shows a sketch of part of the curve with equation ? = 9 3? 5?2 ?3 and the line with equation ? = 4 4?. The line cuts the curve at the points ? 1,8 and ? 1,0 . Find the area of the shaded region between ?? and the curve. 62 3 ? 3 ? ? Find the area of the finite region bounded by the curve with equation ? = 1 ? ? + 3 and the line ? = ? + 3. 4 4.5 ? The diagram shows part of the curve with equation ? = 3 ? ?3+ 4 and the line with equation ? = 4 1 2?. a) Verify that the line and the curve cross at ? 4,2 . b) Find the area of the finite region bounded by the curve and the line. 9 4 ? 7.2 ?
Exercise 11D (Probably more difficult than you d see in an exam paper, but you never know ) The diagram shows a sketch of part of the curve with equation ? = ?2+ 1 and the line with equation ? = 7 ?. Q6 a) Find the area of ?1. b) Find the area of ?2. ? 7 ?1= 205 ?2= 171 ? ?1 6 6 ?2 ? 7
Trapezium Rule Instead of infinitely thin rectangular strips, we might use trapeziums to approximate the area under the curve. y4 y3 What is the area here? y2 ???? =1 2 ?1+ ?2 +1 2 ?2+ ?3 +1 2 ?3+ ?4 y1 ? h h h
Trapezium Rule In general: width of each trapezium ? ??? 2?1+ 2 ?2+ + ?? 1 + ?? ? Area under curve is approximately Example We re approximating the region bounded between ? = 1, ? = 3, the x-axis the curve ? = ?2 x 1 1.5 2 2.5 3 y 1 2.25 4 6.25 9 = 0.5 ???? 8.75 ? ?
Trapezium Rule May 2013 (Retracted) Bro Tip: You can generate table with Casio calcs . ???? 3 (?????). Use Alpha button to key in X within the function. Press = 0.8571 ? ???? =?.? ?.???? + ? ?.???? + ?.???? + ?.???? + ?.???? + ?.???? = ?.??? ? ?
