Calculus: Analytic Geometry and Calculus, with Vectors

266 Integrals 15 A cylindrical solid of radius R has its axis on the x axis of an x, y, z coordi- nate system and has its ends i ...

4.7 Mass, linear density, and moments 267 be describedroughly as a circular ribbon having radius a sin Sk, width ai Ok, length 2 ...

268 Integrals In case 0 < b < a, this gives (11) U-2b2C(b l a)-(a-b)- bb +a2 b-aJ=O and (5) shows that F = 0 as we wish to ...

4.8 Moments and centroids in E2 and E3 269 This answer can be greatly improved if we notice that methods very similar to those w ...

270 Integrals the set of points (x,y) for which a < x S b and fl(x) 5 y < f2(x), then everything is quite simple and f(x) ...

4.8 Moments and centroids in Ez and E, 271 val (or the sum of the lengths of the intervals) in which the line having the equatio ...

272 Integrals fully rigorous about the matter, we assume that the Riemann integral in (4.85) Y = f b 1(x) dx exists and is the v ...

4.8 Moments and centroids in Es and Es 273 In case p = 0, the pth moment is the mass of the body. We shall not comment upon seco ...

274 Integrals Problems 4.89 1 Find the pth moment about the line having the equation x = of the lamina of constant areal density ...

4.8 Moments and centroids in E2 and E3 275 dl < d < d2 and the magnitude of the force is p11, where p is wd, the pressure ...

276 Integrals 8 Using the fact that the pressure at depth d below the surface of water is wd, but without using more formulas fr ...

4.9 Simpson and other approximations to integrals 277 function F for which f f (x) dx = F(x) orf(x) = F' (x) and can then evalua ...

278 Integrals course, we should try to minimize errors by choosing the heights of the rectangles in such a way that, in each str ...

4.9 Simpson and other approximations to integrals 279 in which n is always an evenpositive integer, we begin by approximating f( ...

280 Integrals which appears in (4.95) and is so important that it merits reproduction. We recall that n must be even and that h ...

4.9 Simpson and other approximations to integrals 281 3 Using the Simpson formula with it = 2, obtain the approximations and 4 1 ...

282 Integrals Remark: The last cases show results obtained from an electronic computer that makes 8D calculations. When n is lar ...

4.9 Simpson and other approximations to integrals 283 9 An application of (2) of Problem 8 gives the famous old prismoidal formu ...

Functions, 5 graphs, Functions, graphs, and numbers 5.1 Graphs, slopes, and tangents It is quite possible that we first heard ab ...

5.1 Graphs, slopes, and tangents 285 Figure 5.11 Figure 5.12 Figure 5.13 xo to emphasize the fact that it remains fixed througho ...