Calculus: Analytic Geometry and Calculus, with Vectors

226 Integrals bounded and hence that there is a constant positive M for which -M <_ f (x) S M or I f (x) I<= M. Therefore, ...

4.3 Properties of integrals 227 When problems are being solved, it is always convenient to use the bracket symbol in the formula ...

(^228) Integrals Problems 4.39 (^1) Make a small table of integrals by copying formulas from the second column of (4.171) to (4. ...

4.3 Properties of integrals 229 and, after observing that x log 2 = log 2x and hence f(x) = 2z, put (4) in the form limf(1 - s) ...

(^230) Integrals opinion. Look at Figure 4.392 and note that S seems to fill up about one-third of the square having opposite ve ...

4.3 Properties of integrals 231 and we look at an example. Let A be a positive number and start with the fact that (1) when 0 &l ...

232 Integrals over the interval - oo < x < co, except that (2)fails to hold when n is 1 or 2 and x is an integer. They all ...

4.3 Properties of integrals 233 B6 -47,B7 = 0, B8 = -'96, and that IB2n1 is very large whenn is large. Some books, particularly ...

(^234) Integrals With the aid of these formulas, it is easy to verify (1) for the cases in whichs is 1 and 2 and 3. In fact, Arc ...

4.4 Areas and integrals 235 When (2) holds and u is a differentiable function of x, we can use the chain rule to obtain (3) dxfo ...

236 Integrals escape this awkward situation with the aid of a definition designed for the purpose. Definition 4.42 If R is a rec ...

4.4 Areas and integrals 237 It is possible to describe complicated rules for constructing sets S for which no such number ISI ex ...

(^238) Integrals When an enlightened scientist must calculate the area ISI of S, he writes (4.48) ISI = lim If(,) Ax = f a'f(x) ...

4.4 Areas and integrals 239 two regions R1 and R2. Use partitions and Riemann sums to obtain the formulas IR1I = f 0 -2[fi(x) - ...

240 Integrals 5 Use the technique of the text to find the area of the triangular patch bounded by the lines having the equations ...

4.4 Areas and integrals 241 16 Let .4 be the area of the circular disk of radius a shown in Figure 4.493. Explain the ideas asso ...

(^242) Integrals 20 This problem is interesting because it shows how a basic formula involving areas (a well-known formula which ...

4.4 Areas and integrals 243 and this is equivalent to the relation !6) At Jim 09 = 1 - t2 = 1/1 - sine B = cos 6 ne-.o or to the ...

244 Integrals 24 If f(x) = JxJ, then f(x) = sgn x except when x = 0. When a < 0< b, Theorem 4.37 does not guarantee correc ...

4.5 Volumes and integrals 245 We now illustrate the "slab method" for finding volumes of three- dimensional sets that are common ...