C2 Chapter 11 Integration Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 17th October 2013 Recap 2ðĨ 2 2 3 3 2 + 3ðĨ ððĨ = ðĨ + ðĨ + ? ð 3 2 1+ðĨ+ ðĨ 7 −3 ðĨ2 + ðĨ3 1 2 1 3 1 4 ððĨ = ðĨ + ðĨ + ðĨ ? + ðĨ + ð 2 3 4 3 −4 ððĨ = − ðĨ 3 + ð 4 ? 1 2+ ðĨ −2 −1 ððĨ = −2ðĨ − 2ðĨ ?+ ð 2 ðĨ ðĨ ðĨ 2 −3 ððĨ = − ðĨ 2 + ð 4 ðĨ 3 ? Definite Integration ðĶ Suppose you wanted to find the area under the curve between ðĨ = ð and ðĨ = ð. ð ðŋðĨ ð ðĨ We could add together the area of individual strips, which we want to make as thin as possible… Definite Integration ðĶ ðĶ = ð(ðĨ) ðŋðĨ ðĨ1 ð ðĨ2 ðĨ3 ðĨ4 ðĨ5 ðĨ6 ðĨ ðĨ7 ð What is the total area between ðĨ1 and ðĨ7 ? 7 ð ð ðĨð ðŋðĨ ð=1 ð ðĨ ððĨ As ðŋðĨ → 0 ð Definite Integration ð ð ðĨ ððĨ ð You could think of this as “Sum the values of ð(ðĨ) between ðĨ = ð and ðĨ = ð.” ðĶ Reflecting on above, do you think the following definite integrals would be positive or negative or 0? ðĶ = sin ðĨ ð 2 − ïŧ + ïž 0ïŧ sin ðĨ ððĨ − ïŧ +ïŧ 0 ïž sin ðĨ ððĨ − ïž +ïŧ 0ïŧ sin ðĨ ððĨ 0 ð 2ð ðĨ 2ð 0 2ð ð 2 Evaluating Definite Integrals 2 3ðĨ 2 ððĨ 1 = ðĨ 3? 12 = 23 −? 13 =7 ? ð ð ð ′ ðĨ ððĨ = ð ðĨ ð ð We use square brackets to say that we’ve integrated the function, but we’re yet to involve the limits 1 and 2. Then we find the difference when we sub in our limits. = ð ð − ð(ð) Evaluating Definite Integrals 2 2ðĨ 3 + 2ðĨ ððĨ 1 1 4 = ðĨ + ðĨ2 2 2 1 1 = 8 + 4 −? + 1 2 21 = 2 −1 4ðĨ 3 + 3ðĨ 2 ððĨ −2 = ðĨ 4 + ðĨ 3 −1 −2 = 1 − 1 −? 16 − 8 = −8 Bro Tip: Be careful with your negatives, and use bracketing to avoid errors. Exercise 11B 1 Find the area between the curve with equation ðĶ = ð ðĨ the ðĨ-axis and the lines ðĨ = ð and ðĨ = ð. a ð ðĨ = 3ðĨ 2 − 2ðĨ + 2 c ð ðĨ = ðĨ + 2ðĨ 8 e ð ðĨ = ðĨ3 + ðĨ 2 ð = 0, ð = 2 ð = 1, ð = 2 ð = 1, ð = 4 ? ? ? ð ð. ðð ððð ðð The sketch shows the curve with equation y = ðĨ(ðĨ 2 − 4). Find the area of the shaded region (hint: first find the roots). ð ? 4 6 Find the area of the finite region between the curve with equation ðĶ = (3 − ðĨ)(1 + ðĨ) and the ðĨ-axis. ð ðð? ð Find the area of the finite region between the curve with equation ðĶ = ðĨ 2 2 − ðĨ and the ðĨ-axis. ð ð? ð Harder Examples Find the area bounded between the curve with equation ðĶ = ðĨ 3 − ðĨ and the ðĨ-axis. ðĶ Sketch: (Hint: factorise!) ? −1 1 ðĨ 1 Looking at the sketch, what is −1 ðĨ 3 − ðĨ ððĨ and why? 0, because the positive and negative ? region cancel each other out. What therefore should we do? Find the negative and positive region separately. 0 1 3 1 3 − 3 ððĨ = − 1 ðĨ ðĨ − 3 ððĨ = + −1 0 ð ð So total area is + 4 ð ð = ð ð ? 4 Harder Examples Sketch the curve with equation ðĶ = ðĨ ðĨ − 1 ðĨ + 3 and find the area between the curve and the ðĨ-axis. The Sketch The number crunching ðĨ ðĨ − 1 ðĨ + 3 = ðĨ 3 − 2ðĨ 2 − 3ðĨ ðĶ 0 −3 1 ? -3 1 ðĨ 0 ðĨ 3 − 2ðĨ 2 − 3ðĨ ððĨ = 11.25 ðĨ3 − 2ðĨ 2 7 − 3ðĨ ððĨ = − ? 12 7 5 Adding: 11.25 + 12 = 11 6 Exercise 11C Find the area of the finite region or regions bounded by the curves and the ðĨ-axis. 1 ðĶ =ðĨ ðĨ+2 2 ðĶ = ðĨ+1 ðĨ−4 3 ðĶ = ðĨ+3 ðĨ ðĨ−3 ðĨ2 4 ðĶ= ðĨ−2 5 ðĶ =ðĨ ðĨ−2 ðĨ−5 1 1? 3 5 20 ?6 1 40? 2 1 1? 3 1 21? 12 Curves bound between two lines ðĶ = ð(ðĨ) ðŋðĨ ðĨ ð ð ð Remember that ð ð(ð) meant the sum of all the ðĶ values between ðĨ = ð and ðĨ = ð (by using infinitely thin strips). Curves bound between two lines ð ð ðĨ 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? ð ðĨ − ? ð(ðĨ) ð therefore area… ðī= ð ð ðĨ ?− ð ðĨ Curves bound between two lines Find the area bound between ðĶ = ðĨ and ðĶ = ðĨ 4 − ðĨ . ðĶ 3 ðĨ 4−ðĨ ? − ðĨ ððĨ = 4.5 0 ðĨ Bro Tip: Always do the function of the top line minus the function of the bottom line. That way the difference in the ðĶ 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) ðĨ = −4, ðĨ =?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 ð ðĻððð = ðð? ð Exercise 11D 1 A region is bounded by the line ðĶ = 6 and the curve ðĶ = ðĨ 2 + 2. a) Find the coordinates of the points of intersection. b) Hence find the area of the finite region bounded by ðīðĩ and the curve. 3 The diagram shows a sketch of part of the curve with equation 2 3 ðĶ = 9 − 3ðĨ − 5ðĨ − ðĨ and the line with equation ðĶ = 4 − 4ðĨ. The line cuts the curve at the points ðī −1,8 and ðĩ 1,0 . Find the area of the shaded region between ðīðĩ and the curve. ðī −2,6 ðĩ 2,6 2 ðīððð = 10 3 ? ðī ðĩ 2 3 ? 6 4 Find the area of the finite region bounded by the curve with equation ðĶ = 1 − ðĨ ðĨ + 3 and the line ðĶ = ðĨ + 3. 4.5 ? 9 The diagram shows part of the curve with equation ðĶ = 3 ðĨ − ðĨ 3 + 4 and the line 1 with equation ðĶ = 4 − 2 ðĨ. a) Verify that the line and the curve cross at ðī 4,2 . b) Find the area of the finite region bounded by the curve and the line. 4 ? 7.2 ðī Exercise 11D (Probably more difficult than you’d see in an exam paper, but you never know…) Q6 The diagram shows a sketch of part of the curve with equation ðĶ = ðĨ 2 + 1 and the line with equation ðĶ = 7 − ðĨ. a) Find the area of ð 1 . b) Find the area of ð 2 . ðĶ 7 ð 1 5 6 1 ð 2 = 17 6 ð 1 = 20 ? ð 2 7 ðĨ Trapezium Rule y4 y3 What is the area here? y2 ðīððð 1 = â ðĶ1 + ðĶ2 2 1 ? + â ðĶ2 + ðĶ3 2 1 + â ðĶ3 + ðĶ4 2 y1 h Instead of infinitely thin rectangular strips, we might use trapeziums to approximate the area under the curve. h h Trapezium Rule In general: width of each trapezium ð ð â ðĶ ððĨ ≈ ðĶ1 + 2 ðĶ2 + âŊ + ðĶð−1 + ðĶð 2 Area under curve is approximately Example We’re approximating the region bounded between ðĨ = 1, ðĨ = 3, the x-axis the curve ðĶ = ðĨ 2 x 1 1.5 2 2.5 3 y 1 2.25 4 6.25 9 ? â = 0.5 ? ðīððð ≈ 8.75 Trapezium Rule May 2013 (Retracted) Bro Tip: You can generate table with Casio calcs . ðððð → 3 (ððððð). Use ‘Alpha’ button to key in X within the function. Press = 0.8571 ? ðĻððð = ð. ð ð. ðððð + ð ð. ðððð + ð. ðððð + ð. ? ðððð + ð. ðððð + ð. ðððð = ð. ððð ð To add: When do we underestimate and overestimate?