ACJC 2026 Definite Integrals Lecture Notes

11 DEFINITE INTEGRALS SYLLABUS ▪ Concept of definite integral as a limit of sum - Understand the idea that the area under a curve is a limit of a sum of the areas of rectangles, and use a graphing calculator to illustrate the limiting process (simple cases only) ▪ Definite integral as the area under a curve - Define the definite integral as a limit of sums and interpret it as the area under a curve - Students are expected to know the different perspectives of the concept of definite integral f ( ) d F( ) F( ) b a x x b a=− , where d F( ) f ( )d xxx = (anti-derivative) δ f ( ) d lim δ b a x x x y x → =  , where f ( )yx= (limit of sums) f ( ) d b a xx is the area under the curve f ( )yx= between =xa and =xb ▪ Use anti-derivatives to evaluate definite integrals ▪ Finding the area of a region bounded by a curve and lines parallel to the coordinate axes, between a curve and a line, or between 2 curves ▪ Finding area below the x-axis ▪ Finding the volume of revolution about the x- or y-axis ▪ Finding the approximate value of a definite integral using a graphing calculator or a graphing software

ACJC 2025/26 H2 Mathematics (9758) 2 CONTENTS 1 Definite Integrals .............................................................................. 3 1.1 Evaluating Definite Integrals .................................................. 3 1.2 Definite Integrals Involving Modulus ..................................... 5 2 Plane Area ........................................................................................ 7 2.1 Definite Integral as a Limiting Sum ........................................ 7 2.2 Area Bounded by a Curve and the x-axis .............................. 11 2.3 Area Bounded by a Curve and the y-axis .............................. 13 2.4 Area between Two Curves .................................................... 15 3 Volume of Revolution (Disc Method) ............................................ 18 3.1 Volume of Solid of Revolution about the x-axis................... 18 3.2 Volume of Solid of Revolution about the y-axis................... 20 3.3 Volume of Solid of Revolution Generated by Region between Two Curves ........................................................................... 23 Annex A: Riemann and Riemann Sums ................................................ 28 Annex B: Practice Questions on Definite Integrals ............................... 29

11 Definite Integrals 3 LECTURE 1 Lesson Outline • Evaluating Definite Integrals • Definite Integrals Involving Modulus • Definite Integral as a Limiting Sum 1 DEFINITE INTEGRALS 1.1 Evaluating Definite Integrals A function F( )x is an anti-derivative of a function f ( )x if F ( ) f( ) =xx for all x in the domain of f. If a function f has an anti-derivative F on a x b , then  f ( )d F( ) F( ) F( ). b b aa x x x b a= = − Example 1 Evaluate the following definite integrals exactly. (a) 2 2 1 23dxx x− (b) 0 cos d   x x x Solution (a) 2 223 11 23 d 2ln − = −   x x x xx ( ) ( ) ( ) 23 1 3 2ln 1 2 2 2ln 2 1 2ln1 7 2ln 2 = −    = = − − − = − x x x x x (b)  000 cos d sin sin d   =− x x x x x x x  0cos cos cos0 2  = =  − = −x ■

ACJC 2025/26 H2 Mathematics (9758) 4 Example 2 Find 2 edxxx . Hence find 21 3 0 edxxx exactly. Solution ( ) 2 2 2 f ( ) f ( )11e d 2 e d e using f ( )e d e22 = = + = +   x x x x xx x x x c x x c ( ) ( ) 22 22 2 11 32 00 1 1 2 00 1 0 e d e d 11e (2 ) e d22 11ee22 1 1 1e e 12 2 2 =     =−         =−  = − − =   xx xx x x x x x x x x x Calculator Activity – Using GC to verify answers 1) Press [ALPHA] [WINDOW] and select [4: fnInt( ]. 2) Input 21 3 0 edxxx by filling in the appropriate blanks and press [ENTER]. If exact answers are not required, then all definite integrals should be calculated using the GC. Example 3 By means of the substitution cosxu= , find the exact value of 1 2 0 1d xx− . Solution 1 2 0 1d xx− π 2 cos d sind 0 cos 0 1 cos 1 0 xu x uu x u u x u u = =− =  =  = =  =  = ■

11 Definite Integrals 5 ( ) 0 2 0 2 0 2 2 2 d1 cos d d sin sin d sin d    =− =− =    xuu u u u u uu 0 0 2 2 1 1 cos 2 d2 1 sin 2 22   =− =−   uu uu ( )1 0 0 02 2 4    = − − − = ■ 1.2 Definite Integrals Involving Modulus To evaluate integrals of functions involving modulus, we often will need to split the integrals into parts where the function is positive or negative. This is because f ( ), f ( ) 0f ( ) f ( ), f ( ) 0 = − xxx xx This can be done by considering graph of f ( )=yx or by solving the inequality f ( ) 0x and then using the definition of f ( )x . Example 4 (a) Find 2 2 0 1 dxx − , showing your working clearly. (b) Find the exact value of 1 0 dx x a x− , where 01 a . Solution (a) 2 2 0 1dxx − 12 22 01 1 d 1dx x x x= − + − 1233 0133 1 8 11 2 13 3 3 2 xxxx   = − + −            = − + − − −           = Can we derive the answer without integration? y x 1 2

ACJC 2025/26 H2 Mathematics (9758) 6 (b) 1 0 d , 0 1x x a x a−   ( ) ( ) 1 0 12 3 3 2 0 dd 2 3 3 2 a a a a x a x x x x a x ax x x ax = − + −    = − + −        3 3 3 3 3 1 2 3 3 2 3 2 1 3 2 3 a a a a a aa     = − + − − −         = − + Self-Practice 1 Find the exact value of 2 1 2 1 dxx x− . Solution 2 1 2 1 dxx x− 12 1 12 11 ddx x x xxx= − + − ( ) 1222 1 12 ln | | ln | |22 1 1 1 1ln 2 ln 22 2 8 2 17 8 xxxx   = − + −               = − − − + − −                 = ■ , , x a x axa a x x a −−=  − y x O 1 ■

11 Definite Integrals 7 2 PLANE AREA 2.1 Definite Integral as a Limiting Sum Consider the curve with equation f ( )=yx , where f ( ) 0x for all [ , ]x a b . The area bounded by the curve, the x-axis and the lines xa= and xb= is d f ( )d . bb aa y x x x= Derivation Suppose we divide the required area into vertical strips of equal thickness, say x. Area of a typical strip = δyx , where f ( )=yx Sum of area of all strips = = = xb xa yx Area bounded by the curve and the x-axis from xa= to xb=  = =  xb xa yx As x (thickness of the strips) gets smaller i.e. 0→x , the approximation gets better. It can be shown that total area tends to the actual area of the region bounded by the curve f ( )=yx , the x-axis and the lines xa= and xb= , i.e. 0 lim d = → = =  xb b ax xa y x y x . a b x y O y a b x y

ACJC 2025/26 H2 Mathematics (9758) 8 EXPLORATION / Area under graph as a limit of sums of rectangles Use this GeoGebra applet to visualise the approximation of area under graphs with area of rectangular strips. In the applet, you can change the function, the number of strips and the region. https://www.geogebra.org/m/nKU3jkYd Example 5 The region R is bounded by the curve with equation ln(2 )=+yx , the x- axis and the lines 0=x and 1x = . By considering rectangles of widths 1 n as shown in the diagram, show that the sum of areas of the n rectangles is given by 1 11 ln 2 = − + n r r nn . Using the graphing calculator, estimate the limit of the area of R up to two decimal places. Solution Area of first strip 10ln 2nn =+  Area of second strip 11ln 2nn =+  Area of third strip 12ln 2nn =+  Therefore, it can be seen that area of r-th strip 11ln 2 r nn −=+  Hence, area of all n rectangles = 1 11 ln 2 n r r nn = − + . ■ y x O 1

11 Definite Integrals 9 Calculator Activity – Estimating Limits of Sums (1) Press [Y=]. Type in 1 11 ln 2 X r r XX = − + for Y1. Press [ENTER]. (2) Press [2ND] [WINDOW] to adjust table setup (3)