Chapter Review

Definitions:

• Partition of an interval \([a,b]\) (p. 345)
• Area \(A\) under the graph of a function \(f\) from \(a\) to \(b\) (p. 348)

Definitions:

• Riemann sums (pp. 353-354)
• The definite integral (p. 355)
• \(\int_{a}^{a}f(x)\,dx=0\) (p. 356)
• \(\int_{a}^{b}f(x)\,dx=-\int_{b}^{a}f(x)\,dx\) (p. 356)

Theorems:

• If a function \(f\) is continuous on a closed interval \([a,b]\), then the definite integral\( \int_{a}^{b}f(x)\,dx\) exists. (p. 356)
• \(\int^b_a h\ dx = h(b-a)\), \(h\) a constant (p. 357)

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

• Part 1: The function \(I\) defined by \(I\,(x)=\int_{a}^{x}f(t)\,dt\) has the properties that it is continuous on \([a,b] \) and differentiable on \((a,b).\) Moreover, \(I^\prime (x)=\dfrac{d}{dx} \left[ \int_{a}^{x}f(t)\,dt\right] =f(x),\) for all \(x\) in \((a,b).\) (p. 362)
• Part 2: If \(F\) is any antiderivative of \(f\) on [\(a,b\)], then \(\int_{a}^{b}f(x)\,dx=F(b)-F(a)\). (p. 364)

Properties of definite integrals:

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

• Integral of a sum: \(\int_{a}^{b}\,[ f(x)+g(x)] \,dx=\int_{a}^{b}f(x)\,dx + \int_{a}^{b}g(x)\,dx \) (p. 369)
• Integral of a constant times a function: \(\int_{a}^{b}kf(x)\,dx = k\int_{a}^{b}f(x)\,dx \)  (p. 369)
• \(\int_{a}^{b}\left[ k_{{\rm 1}}f_{{\rm 1}}(x)+k_{{\rm 2}}f_{{\rm 2}}(x)+\cdots +\,k_{n} f_{n}(x)\right] \, dx = k_{1}\int_{a}^{b} f_{1}(x)\,dx + k_{2}\int_{a}^{b}f_{2}(x)\,dx+\cdots + k_{n}\int_{a}^{b}f_{n}(x)\,dx \)  (p. 370)
• If \(f\) is continuous on an interval containing the numbers \(a\), \(b\), and \(c\), then \(\int_{a}^{b}f(x)\,dx=\int_{a}^{c}f(x)\,dx+\int_{c}^{b}f(x)\,dx \) 
  (p. 370)
• Bounds on an Integral: If a function \(f\) is continuous on a closed interval \([a,b]\) and if \(m\) and \(M\) denote the absolute minimum and absolute maximum values, respectively, of \(f\) on \([a,b]\), then \(m(b-a)\leq \int_{a}^{b}f(x)\,dx\leq M\,(b-a) \)  (p. 371)
• Mean Value Theorem for Integrals: If a function \(f\) is continuous on a closed interval \([a,b]\), then there is a real number \(u\), where \(a\,\leq \,u\,\leq \,b\), for which \(\int_{a}^{b}f(x)\,dx = f(u)(b-a) \)  (p. 372)

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)

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:

• Derivative of an integral: \(\dfrac{d}{dx}\left[ \int f(x)\,dx\right]=f(x) \)  (p. 380)
• Integral of a sum: \(\int\, [ f(x)+g(x)] dx=\int f(x)\,dx +\int g(x)dx \)  (p. 381)
• Integral of a constant \(k\) times a function: \(\int k\,f(x)\,dx =k\int f(x)\,dx \)  (p. 381)

Method of substitution: (p. 388)

Method of substitution (definite integrals):

• Find the related indefinite integral using substitution. Then use the Fundamental Theorem of Calculus. (p. 391)
• Find the definite integral directly by making a substitution in the integrand and using the substitution to change the limits of integration. (p. 391)

Basic integration formulas:

• \(\int \dfrac{g^\prime (x) }{g(x) }dx\) \(=\ln \vert g(x) \vert +C\)  (p. 390)
• If \(f\) is an even function, then \(\int_{-a}^{a}f(x)\,dx=2\int_{{0}}^{a}f(x)\,dx \)  (p. 393)
• If \(f\) is an odd function, then \({\int_{-a}^{a}}f(x)\,dx=0\).  (p. 393)