Calculus: Analytic Geometry and Calculus, with Vectors

(^246) Integrals perpendicular to the y axis intersect the solid in plane sets the areas ()f which are not easily found. Finally ...

4.5 Volumes and integrals 247 val -a S x 5 a or to take double the result of partitioning the interval 0 5 x 5 a. Remark: Scient ...

248 Integrals 10 Find, in two or three different ways, the volume of the solid obtained by replacing the disk of the preceding p ...

4.5 Volumes and integrals 249 (x,y,z) for which y 0, we make a partition of the interval0 < y 5 b. When 0 < y < b, we c ...

250 Integrals and notations have their historical origins in primitive ideas that are fuzzy or incorrect. The number in the righ ...

4.6 Riemann-Cauchy integrals and work 251 exists as a Riemann integral whenever h? a and that this integral, as a function of h, ...

252 Integrals are defined by these formulas whenever the integrals on the right exist as Riemann-Cauchy integrals. Perhaps atten ...

4.6 Riemann-Cauchy integrals and work 253 except when x = 0 fi and cheerfully make the calculation (4.637) i2 dx=-11 = -2 (????) ...

254 Integrals norm JP1 and write f(xk) Oxk as an approximation to the amount of work done in pulling the particle from the left ...

4.6 Riemann-Cauchy integrals and work It follows from theseformulas that (4.673) lim Wa,b = r°° k=1 dx b.. X a 25S This formula ...

256 01 1 Figure 4.691 Integrals 4 Remembering that e = 2.71828, and remembering or learning that e3 is about 20,e° is about 400, ...

4.6 Riemann-Cauchy integrals and work 257 Figure 4.692,which shows the rod before and after stretching, may be helpful. Supposin ...

(^258) Integrals From this we conclude that there must be a constant vector cl such that (2) mxi = TCt2i. But (dx/dt)i is the ve ...

4.7 Mass, linear density, and moments 259 as the definitionof the work done by F over the time interval a < t < b. Show th ...

260 Integrals monotone increasing over the interval. This means that F(xi) < F(x2) whenever a 5 x1 < x2 < b. Such funct ...

4.7 Mass, linear density, and moments 261 subtracting the total mass in the interval a < x < xk_l. Thus it is the total ma ...

262 Integrals Problems 4.79 1 As suggested by Figure 4.791, let a rod having constant linear density (mass per unit length) 6 be ...

4.7 Mass, linear density, and moments 263 7 When M = 7, the graphs of the functions in the preceding problem are different from ...

264 Integral$ unit vector in its direction. Our next step is to write PPk in terms of the coordi. nates of P and P, and to write ...

4.7 Mass, linear density, and moments 265 concentrated at the point (c,0,0). Hint: Assuggested by Figure 4.793, make a partition ...