Calculus: Analytic Geometry and Calculus, with Vectors

706 Iterated and multiple integrals nates p, ,, z is 3(p,4,z). Set up a threefold iterated integral in polar coordinates for the ...

13.8 Triple integrals; spherical coordinates 707 where G is the, gravitational constant and the last factor isa unit vector in t ...

708 Iterated and multiple integrals can advantageously be expressed in terms of spherical coordinates r, 0, 8. When we use spher ...

13.8 Triple integrals; spherical coordinates 709 this triple integral as an iterated integral. When limits of integration for it ...

710 Iterated and multiple integrals order of integration in such a way that the last integration is with respect to r and simpli ...

13.8 Triple integrals; spherical coordinates 711 to obtain the formula (5) L = f b.Jr2sin20 (dl) 2 + r2(dl)2+ (dr)2 a V dt at dt ...

712 Iterated and multiple integrals It is supposed that the sphere has density 5(r,4,9) at the point P having spherical coordina ...

13.8 Triple integrals; spherical coordinates 713 partitions of the ball and principles of the integral calculusto obtain (6) F = ...

714 Iterated and multiple integrals sin 0 dO and, except for algebraic sign, the integrand becomes that in (14) f (r) =fr D + u ...

Appendix i- Proofs of basic theorems on limits This appendix contains proofs of the basic theorems on limits which were given wi ...

716 Appendix I The first basic theorem shows us that there can be at most one number L for which lim f (x) = L. Theorem A (2) If ...

Proofs of basic theorems on limits 717 Theorem C (8) lim x = a. This theorem tells us that if f(x) = x, then (9) lim f(x) = a. T ...

718 Let a and ei and e2 be positive numbers. S1 and S2 such that Appendix 1 Choose positive numbers (17) I f(x) - LI < ei (0 ...

Proofs of basic theorems on limits 719 whenever 0 < Ix - at < S. Therefore, (28) 1 -.1 i _ M - g(x) < e2 2e2 g(x) MR ' ...

720 Appendix I of ham (or a fly) is between them, then the thing that is caught in the middle must also be near Minneapolis. To ...

Appendix 2 Volumes This appendix involves volumes of sets in E3. Its purpose is to show that the theory of volumes is not simple ...

722 Appendix 2 Figure A.1 shows three spherical shells the inner and outer radii of which are 1 and 2. A point P lies in one of ...

Volumes 723 that (B,), (B2), and (B3) are valid, a contradiction arises from the assump- tion that each bounded set in Ea has a ...

The Greek Alphabet Letters Names Letters Names Letters Names A a alpha I L iota P p rho B 0 beta K K kappa E or sigma r y gamma ...

Index 725 Arithmetico-geometric mean, 343 Astroid (see Hypocycloids, of four cusps) Asymptotes, 135 of hyperbolas, 381 Asymptoti ...