Calculus: Analytic Geometry and Calculus, with Vectors

(^646) Series In this formula we substitute the expression for log n! given in (7) and the expres- sion for the last sum given i ...

12.6 Euler-Maclaurin summation formulas 647 and more formulas can be produced by giving greater values to m. While these formula ...

(^648) Series and k are quite small, it is advisable to substitute in (3) and simplify the result before making numerical calcul ...

12.6 Euler-Maclaurin summation formulas 649 that there are constants M and Dnx such that IDnxj 5 M and (14) log Zn (k/ =log In - ...

650 x = t/2 then gives the formula (21) lim Zn (k) = 1 f A n-a= 2 4 2 n Series The right side of (21) can be evaluated sith the ...

12.6 Euler-Maclaurin summation formulas 651 in which the star on the Z means that the term for which is = 1 is omitted from the ...

13 Iterated and multiple integrals 13.1 Iterated integrals When we differentiate a function f having values f(x) and then iterat ...

13.1 Iterated integrals 653 Substituting this in the formula for f2(x) then gives (13.12) .12(x) = fax (fatf(ti) dti) dt. Replac ...

654 Iterated and multiple integrals insist on the other hand that we should "work from right to left,"so that f c goes with dy a ...

13.1 Iterated integrals 655 3 By evaluating all of the integrals involved, show that Jot dz fozf(Y) dy= Jo` (t - y)f(y) dy when ...

656 Iterated and multiple integrals 10 Supposing that n > -2 and is 76 1, show that 1 x f - 1 dx fo (x I Y)" dY = 2n+1 o (n + ...

13.1 Iterated integrals 657 The integral in (1) will exist as a Riemann-Cauchy integral and will have the value Y if (6) V=Tlim ...

658 Iterated and multiple integrals segments from (xo,yo,zo) to (x,yo,zo) and then to (x,y,zo) and then to (x,y,z) is (6) Wa(x,y ...

13.2 Iterated integrals and volumes 659 Ans.: In each case the answer is u(x,y), and itmay be worthwhile to try to under- stand ...

660 Iterated and multiple integrals to existence of the integrals involved, we investigate the number I defined by (13.22) I = 0 ...

13.2 Iterated integrals and volumes 661 the parts covering Si, S2, S, have heights Z, B, C. We should know that we can undertake ...

(^662) Iterated and multiple integrals and we have shown that, in the special cases being considered,our iterated integral is th ...

13.2 Iterated integrals and volumes 663 without benefit of elegant figures, we can use Figure 13.27 to help us see what we are d ...

664 Iterated and multiple integrals a set S which is a subset of the rectangular set R containing points (x,y) for which a < ...

13.2 Iterated integrals and volumes 665 find a region R such that the formula is valid whenever f(x,y) is continuous over R and ...