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C2 Chapter 11 Integration Dr J Frost ([email protected]) Last modified: 17th October 2013 Recap 2 2 2 3 3 2 + 3 = + + ? 3 2 1++ 7 −3 2 + 3 1 2 1 3 1 4 = + + ? + + 2 3 4 3 −4 = − 3 + 4 ? 1 2+ −2 −1 = −2 − 2 ?+ 2 2 −3 = − 2 + 4 3 ? 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 =1 As → 0 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 2 − + 0 sin − + 0 sin − + 0 sin 0 2 2 0 2 2 Evaluating Definite Integrals 2 3 2 1 = 3? 12 = 23 −? 13 =7 ? ′ = 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. = − () Evaluating Definite Integrals 2 2 3 + 2 1 1 4 = + 2 2 2 1 1 = 8 + 4 −? + 1 2 21 = 2 −1 4 3 + 3 2 −2 = 4 + 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 = the -axis and the lines = and = . a = 3 2 − 2 + 2 c = + 2 8 e = 3 + 2 = 0, = 2 = 1, = 2 = 1, = 4 ? ? ? . The sketch shows the curve with equation y = ( 2 − 4). Find the area of the shaded region (hint: first find the roots). ? 4 6 Find the area of the finite region between the curve with equation = (3 − )(1 + ) and the -axis. ? Find the area of the finite region between the curve with equation = 2 2 − and the -axis. ? Harder Examples Find the area bounded between the curve with equation = 3 − and the -axis. Sketch: (Hint: factorise!) ? −1 1 1 Looking at the sketch, what is −1 3 − 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 = − 1 − 3 = + −1 0 So total area is + 4 = ? 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 1 ? -3 1 0 3 − 2 2 − 3 = 11.25 3 − 2 2 7 − 3 = − ? 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 -axis. 1 = +2 2 = +1 −4 3 = +3 −3 2 4 = −2 5 = −2 −5 1 1? 3 5 20 ?6 1 40? 2 1 1? 3 1 21? 12 Curves bound between two lines = () Remember that () meant the sum of all the values 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 1 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. 3 The diagram shows a sketch of part of the curve with equation 2 3 = 9 − 3 − 5 − 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. −2,6 2,6 2 = 10 3 ? 2 3 ? 6 4 Find the area of the finite region bounded by the curve with equation = 1 − + 3 and the line = + 3. 4.5 ? 9 The diagram shows part of the curve with equation = 3 − 3 + 4 and the line 1 with equation = 4 − 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. 4 ? 7.2 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 = 2 + 1 and the line with equation = 7 − . a) Find the area of 1 . b) Find the area of 2 . 7 1 5 6 1 2 = 17 6 1 = 20 ? 2 7 Trapezium Rule y4 y3 What is the area here? y2 1 = ℎ 1 + 2 2 1 ? + ℎ 2 + 3 2 1 + ℎ 3 + 4 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 ℎ ≈ 1 + 2 2 + ⋯ + −1 + 2 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 ? ℎ = 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 ? = . . + . + . + . ? + . + . = . To add: When do we underestimate and overestimate?