Intergration
LECTURE 2 CALCULUS
INTEGRATION
2.3 INTEGRATION References • • • Barnett, Ziegeler and Byleen 2000a, Chapters 6 and 7. Chiang and Wainwright, Chapter 14. Sydsaeter and Hammond, Chapter 9.
Antiderivatives and Indefinite Integrals Many operations in mathematics have reverses – compare addition and subtraction, multiplication and division, and powers and roots. The reverse of differentiation is antidifferentiation. A function F is an antiderivative of a function f if . , , then the functions differ ,
If the derivatives of two functions are equal on an open interval by at most a constant.
Symbolically: If F and G are differentiable functions on the interval for all x in , , then , for some constant k. We use the symbol
and
called the indefinite integral, to represent the family of all antiderivatives of if The symbol is called an integral sign, and the function .
, and write
is called the integrand.
The symbol dx indicates that the antidifferentiation is performed with respect to the variable x. The arbitrary constant C is called the constant of integration.
1
The fact that indefinite integration and differentiation are reverse operations, except for the addition of the constant of integration, can be expressed symbolically as i.e. the derivative of the indefinite integral of and i.e. the indefinite integral of the derivative of . is is .
Indefinite Integral Formulas and Properties For k and C constants: 1. 2. 3. 4. 5. 6. ln| | . , 0. . , . . 1.
Example 19 Find each indefinite integral. a) b) 6 . .
2
Solution to Example 19 a) b) 6 6 8 = 4 6 8 . 8 6 8 4 6 4 4 . 8
Practice Exercise 19 Find each indefinite integral. a) b) 8 . .
Example 20 The marginal average cost for producing x digital sports watches is given by
,
100
25,
where
is the average cost in Kenya Shillings.
Find the average cost function.
3
Solution to Example 20 Recall that marginal average cost is the derivative of the average cost function. We therefore find the
INTEGRATION
2.3 INTEGRATION References • • • Barnett, Ziegeler and Byleen 2000a, Chapters 6 and 7. Chiang and Wainwright, Chapter 14. Sydsaeter and Hammond, Chapter 9.
Antiderivatives and Indefinite Integrals Many operations in mathematics have reverses – compare addition and subtraction, multiplication and division, and powers and roots. The reverse of differentiation is antidifferentiation. A function F is an antiderivative of a function f if . , , then the functions differ ,
If the derivatives of two functions are equal on an open interval by at most a constant.
Symbolically: If F and G are differentiable functions on the interval for all x in , , then , for some constant k. We use the symbol
and
called the indefinite integral, to represent the family of all antiderivatives of if The symbol is called an integral sign, and the function .
, and write
is called the integrand.
The symbol dx indicates that the antidifferentiation is performed with respect to the variable x. The arbitrary constant C is called the constant of integration.
1
The fact that indefinite integration and differentiation are reverse operations, except for the addition of the constant of integration, can be expressed symbolically as i.e. the derivative of the indefinite integral of and i.e. the indefinite integral of the derivative of . is is .
Indefinite Integral Formulas and Properties For k and C constants: 1. 2. 3. 4. 5. 6. ln| | . , 0. . , . . 1.
Example 19 Find each indefinite integral. a) b) 6 . .
2
Solution to Example 19 a) b) 6 6 8 = 4 6 8 . 8 6 8 4 6 4 4 . 8
Practice Exercise 19 Find each indefinite integral. a) b) 8 . .
Example 20 The marginal average cost for producing x digital sports watches is given by
,
100
25,
where
is the average cost in Kenya Shillings.
Find the average cost function.
3
Solution to Example 20 Recall that marginal average cost is the derivative of the average cost function. We therefore find the