Chapter Review

THINGS TO KNOW

5.1 Area

Definitions:

5.2 The Definite Integral

Definitions:

Theorems:

5.3 The Fundamental Theorem of Calculus

Fundamental Theorem of Calculus: Let \(f\) be a function that is continuous on a closed interval \([a,b]\).

5.4 Properties of the Definite Integral

Properties of definite integrals:

If two functions \(f\) and \(g\) are continuous on the closed interval \([a,b]\) and \(k\) is a constant, then

Definition: The average value of a function over an interval \([a,b] \) is \(\bar{y}=\dfrac{1}{b-a}\int_{a}^{b}f(x)\,dx\)  (p. 374)

5.5 The Indefinite Integral; Growth and Decay Models

The indefinite integral of \(f\): \(\int f(x)\, dx=F(x) +C\) if and only if \(\dfrac{d}{dx}[ F(x) +C] =f(x),\) where \(C\) is the constant of integration. (p. 379)

Basic integration formulas: See Table 1.  (p. 380)

Properties of indefinite integrals:

5.6 Method of Substitution; Newton's Law of Cooling

Method of substitution: (p. 388)

Method of substitution (definite integrals):

Basic integration formulas:

401

OBJECTIVES

Section You should be able to … Examples Review Exercises
5.1 1 Approximate the area under the graph of a function (p. 344) 1, 2 1, 2
2 Find the area under the graph of a function (p. 348) 3, 4 3, 4
5.2 1 Define a definite integral as the limit of Riemann sums (p. 353) 1, 2 5(a), (b)
2 Find a definite integral using the limit of Riemann sums (p. 356) 3–5 5(c)
5.3 1 Use Part 1 of the Fundamental Theorem of Calculus (p. 363) 1–3 7-10, 52, 53
2 Use Part 2 of the Fundamental Theorem of Calculus (p. 365) 4, 5 5(d), 11–13, 15–18, 56
3 Interpret an integral using Part 2 of the Fundamental Theorem of Calculus (p. 365) 6 6–21, 22, 57
5.4 1 Use properties of the definite integral (p. 369) 1–6 23, 24, 27, 28, 53
2 Work with the Mean Value Theorem for Integrals (p. 372) 7 29, 30
3 Find the average value of a function (p. 373) 8 31–34
5.5 1 Find indefinite integrals (p. 379) 1 14
2 Use properties of indefinite integrals (p. 380) 2, 3 19, 20, 35, 36
3 Solve differential equations involving growth and decay (p. 382) 4, 5 37, 38, 59, 60
5.6 1 Find an indefinite integral using substitution (p. 387) 1–5 39–41, 44, 45, 48, 51
2 Find a definite integral using substitution (p. 391) 6, 7 42, 43, 46, 47, 50, 51, 54, 55, 49
3 Integrate even and odd functions (p. 393) 8, 9 25–26
4 Solve differential equations: Newton's Law of Cooling (p. 394) 10 58