The Chemistry Maths Book, Second Edition

130 Chapter 5Integration is the operator form of equation (5.2). The operatorsDandD − 1 have the property D − 1 D 1 = 1 D D − 1 ...

5.2 The indefinite integral 131 EXAMPLE 5.2Find subject to conditions (i)(x, 1 y) 1 = 1 (2, 1 0), (ii)y 1 = 110 whenx 1 = 13. We ...

132 Chapter 5Integration 5.3 The definite integral The integral calculus was invented to solve the problem of finding the area e ...

5.3 The definite integral 133 In the general case, wheny 1 = 1 f(x)is not necessarily a linear function, the integral calculus ( ...

134 Chapter 5Integration (iv) (v) (vi) (vii) 0 Exercises 16–25 Average value of a function Because the definite integral is iden ...

5.3 The definite integral 135 2.If cis a third limit of integration, not necessarily between aand b: (5.15) This is true because ...

136 Chapter 5Integration Figure 5.6 shows that the integrand, sin 1 x, has positive values when 01 < 1 x 1 < 1 π, and nega ...

5.3 The definite integral 137 EXAMPLE 5.7The function is continuous at x 1 = 10 , but not smooth, the slope of its graph changin ...

138 Chapter 5Integration EXAMPLES 5.8Improper integrals (i) The function is not defined at x 1 = 1 0 and the definite integral b ...

5.3 The definite integral 139 For example, using the property of the exponential thate −b 1 → 10 asb 1 → 1 ∞. In practice, infin ...

140 Chapter 5Integration In this case the value of sin 1 boscillates between + 1 and − 1 as bincreases, and no unique value can ...

5.3 The definite integral 141 In general, an arbitrary function is neither even nor odd;f(−x) 1 ≠ 1 ±f(x). It is always possible ...

142 Chapter 5Integration EXAMPLE 5.10 (i) Find the even and odd components off(x) 1 = 1 e x , (ii) evaluate the definite integra ...

5.4 The integral calculus 143 An estimate of the area is then obtained by replacing each strip by a rectangle. The width of the ...

144 Chapter 5Integration Following Leibniz, this limit is written as (5.33) and is called the definite integral of the functionf ...

5.4 The integral calculus 145 The integral is evaluated in Example 6.10; , and the area of the circle is four times this. Method ...

146 Chapter 5Integration The integral as a length Lety 1 = 1 f(x)be continuous in the rangea 1 ≤ 1 x 1 ≤ 1 b, and let sbe the le ...

5.5 Uses of the integral calculus 147 (four times the length of that quarter in the first quadrant, see Example 5.11). The integ ...

148 Chapter 5Integration 5.6 Static properties of matter Consider a set of Ndiscrete massesm 1 ,m 2 , =,m N distributed along a ...

5.6 Static properties of matter 149 where is the total force acting on the system of masses. The position Xof the centre of mass ...