Calculus: Analytic Geometry and Calculus, with Vectors

206 Integrals Thorough understanding of this particular example is of utmost impor- tance because it involves an idea that is ve ...

4.1 Indefinite integrals 207 refer to tables in books of tables in preference to tables in calculus text- books. Teachers can be ...

(^208) Integrals 5 Look at the integral f(1 + 5x)dx and tell what must be done to enable us to evaluate the integral by formula ...

4.1 Indefinite integrals 209 example, find the solutions of the following differential equations satisfying the given boundary c ...

210 This gives the first and hence the second of the formulas e kxy = f1, y = Aekx. Integrals More complete treatments of these ...

4.1 Indefinite integrals 211 Let x be confined to an interval I over which two given functions u and v are differentiable. The s ...

(^212) Integrals solve one part, pick an integral formula from a (preferably your) bool, of tables that has the form f f (x) dx ...

4.2 Riemann sums and integrals 213 subinterval so that to < tl < ti, let t2 be in the second subinterval so that ti <= ...

214 Integrals of using i leads to awkwardness when we finish study of calculus and enter realms where i is always the imaginary ...

4.2 Riemann sums and integrals 215 does not exist, we look briefly at an example. Let f be the dizzy dancer function D, defined ...

216 Integrals finite collection, that is, it may contain only 1 or 2 or 3 or 416 or 31,690 or some other positive integer number ...

4.2 Riemann sums and integrals 217 So far, the integral in (4.271) has been defined only when x > a. now completethe definiti ...

(^218) Integrals by abandoning the good old idea that the elementary functions (polynomials, trigonometric functions, etcetera) ...

4.2 Riemann sums and integrals 219 7 This problem requires us to attain a complete understanding of a more complex situation. Le ...

220 Integrals 10 In a campaign to obtain good ideas about Riemann sums and integrals, we can use the discontinuous function 4, d ...

4.2 Riemann sums and integrals 221 12 Our purpose is to discover that some very obvious and superficially useless remarks about ...

(^222) Integrals and (7) n)-1 j n LI J (tk) Otk = I f(pu + q)p Auk. k=1 k-1 The result (1) follows from this. Let I denote the l ...

4.2 Riemann sums and integrals 223 14 Assuming that the integrals exist, show that (1) f hf(x) dx= fohf(-x) dx. Remark: This inn ...

224 Integrals 4.3 Properties of integrals In what follows, all integrals bearing limits of integration are Riemann integrals. Th ...

4.3 Properties of integrals 225 Theorem 4.341 If k is a constant, then f-22 k dt = k(x2 - xl). If a < b, if f is integrableov ...