TMJC H2 Chapter 11 Definite Integrals Learning Package 2024

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Chapter 11 Definite Integrals TMJC 2024 Page 1 of 33 H2 Mathematics (9758) Chapter 11 Definite Integrals Core Concept Notes Success Criteria: Surface Learning Deep Learning Transfer Learning  Evaluate definite integral using anti- derivatives i.e. f ( ) d F( ) F( ) b a x x a b=− , where d F( ) f ( )d xxx = (anti-derivative).  Evaluate the approximate values of definite integrals using a graphing calculator.  Interpret definite integrals as the area under a curve.  Understand that definite integral is negative when the curve is below the x-axis.  Evaluate an integrand of the form f ( )x without the use of GC.  Find the area of a region bounded by a curve and lines parallel to the coordinate axes.  Find the area of a region between a curve and a line.  Find the area of a region between two curves.  Find the volume of revolution about the x- or y-axis when it is bounded by one curve.  Explain that the area under a curve is the limit of a sum of the areas of rectangles and relate definite integral as a limit of sums i.e. 0 f ( ) d lim , b xa x x y x → =    where f ( )yx= .  Use a graphing calculator to illustrate the limiting process for simple cases.  Find the volume of revolution about the x- or y-axis when it is bounded by two curves. Definition of Learning (John Hattie): The process of developing sufficient surface knowledge to then move to deeper understanding such that one can appropriately transfer this learning to new tasks and situations.

Chapter 11 Definite Integrals TMJC 2024 Page 2 of 33 y x y = x2 1 2 §1 Definite Integral is a Process of Summation 1.1 Definite Integral as the Limit of a Sum Suppose we wish to find the area bounded by the curve 2yx= , the x-axis, the lines x = 1 and x = 2 as shown in Figure 1. Let us estimate this area by dividing it into 5, 20 and 50 equal rectangular strips as shown in the figures below and finding the total area of the rectangular strips. When n = 5 When n = 20 Total area of the rectangles = 2.04 units2 Total area of the rectangles = 2.25875 units2 1 2 y = x2 y x Figure 1 y x y = x2 O 1 2 Please watch the video in SLS before this segment: You may refer to Annex A for a more comprehensive explanation.

Chapter 11 Definite Integrals TMJC 2024 Page 3 of 33 When n = 50 From the diagrams above, the total area of the rectangular strips gives an approximate value for the required area. Observe that increasing the number of rectangular strips gives better approximation to the required area. When n tends to infinity Total area of the rectangles = 2.3034 units2 y x y = x2 2 1 Explanation: Let ( ) 2f. xx = Using integration, the actual area bounded by 2yx= , 1x = , 2x = and the x-axis is given by ( ) 232 2 1 1 7 d or 2.33 (3s.f)33 xxx == . Also, the same area can be estimated by first, dividing x values from 1 to 2 into n equal parts, forming n rectangles of equal width of 1 n . Hence, the same area is estimated by finding the total areas of the rectangles ( ) ( )1f 1 f 1 f 11 1 1 1 1 2 ... f 1 11nn n n n n n n           =  +  + +  + + +  + −                     As the number of rectangles, n, increases to infinity, ( ) ( ) ( ) ( ) 2 1 lim f 1 f 1 f 1 2 ... f 1 1 f d 2.33 3 s.f1 1 1 1 n n x xn n n n→            + + + + + + + − = =                       y x y = x2 2 1 … When n tends to infinity, the total area of the rectangles will approach the actual area bounded by the graph 2yx= and 1x = , 2x = and the x-axis, which is ( ) 2 2 1 d 2.33 3 s.fxx =

Chapter 11 Definite Integrals TMJC 2024 Page 4 of 33 Example 1 [N2001/1/18] (i) Find the exact value of 1 2 0 1 1 dxx+ . (ii) The graph of 2 1 1y x= + , for 01 x , is shown in the diagram. Rectangles, each of width 1 n , are drawn under the curve. Show that the total area A of all n rectangles is given by 222 1 1 1 1 1 ... 21 2 3111 A n nnn  = + + + +      +++            . State the limit of A as n → . (i) 1 2 0 1 d1 xx+ 11 0 tan x−=  π 4= 1 O y x 1

Chapter 11 Definite Integrals TMJC 2024 Page 5 of 33 (ii) To find the height of each rectangle, let 1 2 3 1, , ,..., ,1 nx n n n n −= . 2 2 2 1 .. 1 . 1 111 11 1 22 11 1 3n n A nn n nn       +      +       +   +    =+  + ( )222 1 1 1 1 1 ... shown21 2 3111 A n nnn  = + + + +      +++            As n → , sum of area of all n rectangles will tend to area under graph. πlim 4n A → = 1 O y x 1 𝑦 = 1 1 + 𝑥2 2 1 41 y n = +  ( ) 2 1 11 y = + 2 1 21 y n = +  2 1 11 y n = + 

Chapter 11 Definite Integrals TMJC 2024 Page 6 of 33 §2 Area under a Curve 2.1 Area Bounded by a Curve and the x-axis (I) Area Above the x-axis The curve f ( )yx= is continuous on the interval bxa  . The area bounded by the curve ( )xy f= where ( )f0 x  , the x-axis, the lines ax = and bx = , A, is given by Example 2 [2014/JJC Prelim/I/4 modified] The diagram shows a region R in the first quadrant bounded by the curve C with equation 2 2 3 4 y x =+ − , the x-axis, the y-axis and the line 3x = . Calculate the exact area of region R. Area = 3 0 dy x 3 20 2 3d 4 x x =+ − 3 1 0 2sin 3 2 x x− =+   ( ) 11 302sin 3 3 2sin 3 022 −−   = + − +     2 333 =+ x y O R C x = 2 d b a A y x=  a b y=f(x) y x O Observation 2 2 4 x−: is constant quadratic  MF27 MF27 Pg 4 1 22 1 d sin xx aax − = −   Note that exact answer is required, which means detailed working of integration must be shown. Good Strategy: Check your final answer using GC (Pg 7) Region R is bounded by C and x-axis from 0x = to 3x =

Chapter 11 Definite Integrals TMJC 2024 Page 7 of 33 Method 1 : Using GC Home Screen (Recommended when the quadrant the area lies in is known) Steps/Keystrokes/Explanations Screen Display 1. Press a@ and select 4: fnInt(. 2. Key in the lower and upper limits, integrand and variable of integration and press e Method 2: Using GC Graph Screen Steps/ Keystrokes/ Explanations Screen Display 1. Press ! to graph the function Y1 = 2 2 3 4 x + − . 2. Press @ to set the settings as shown on the screen. This is to have a better view of the graph. Basic knowledge of y = 2 2 3 4 x + − is required. 3. Press % to view th