SAJC 2022 Chapter 8 Apps of Integration (Tutor)

SAJC 2022 JC2 H1 Mathematics Chapter 8: Applications of Integration Page 1 of 20 Chapter 8 (Pure Mathematics): Applications of Integration Objectives: At the end of the chapter, you should be able to: (a) evaluate definite integrals (b) find the area of a region bounded by a curve and lines parallel to the coordinate axes, between a curve and a line, or between two curves (c) find the numerical value of a definite integral using a graphing calculator Content 8.1 What is a Definite Integral 8.2 Evaluate Definite Integrals 8.2.1 Evaluating Definite Integral Algebraically 8.2.2 Evaluating Definite Integral Using Graphing Calculator (GC) 8.3 Basic Properties of Definite Integrals 8.4 Area Under a Curve 8.4.1 Area under the curve with respect to x-axis 8.4.2 Area between the curve and y-axis 8.4.3 Area between two curves References 1. New Additional Mathematics, Ho Soo Thong (Msc, Dip Ed), Khor Nyak Hiong (Bsc, Dip Ed) 2. New Syllabus Additional Mathematics (7th Edition), Shinglee Publishers Ptd Ltd

SAJC 2022 JC2 H1 Mathematics Chapter 8: Applications of Integration Page 2 of 20 8.1 What is a Definite Integral In the previous chapter you learnt that integration is the reverse of differentiation but what does the integral function do? In this chapter, you will learn that you can use the integral to find the area of a function over an interval [a, b] where a is called the lower limit and b is called the upper limit. Area under the curve over the interval [a, b] =  f ( ) d F( ) F( ) F( ) b b aa x x x b a= = − . We will go into more details in Section 8.4. 8.2 Evaluate Definite Integrals Steps to find a definite integral 1. Integrate the function. 2. Substitute the upper limit, b, into x to obtain F( )b . 3. Substitute the lower limit, a, into x to obtain F( )a . 4. F( ) F( )ba− . The answer is the area under the curve. 8.2.1 Evaluating Definite Integral Algebraically Example 1 Evaluate (a) 2 3 1 x dx (b) ( ) 2 3 0 25x dx − + . Solution: (a) 242 3 1 1 24 1 44 d 4 1 4 1 214 15 3 344 xxx x = =  =−  ==  a b x y y = f(x) Step 1: Integrate the function 3x to get 4 4 x . Step 2: Substitute the upper limit, 2, into x to obtain 42 4 . Step 3: Substitute the lower limit, 1, into x to obtain. 41 4 Step 4: 4421 44− . The answer is the area under the curve.

SAJC 2022 JC2 H1 Mathematics Chapter 8: Applications of Integration Page 3 of 20 (b) ( )