. Applied calculus; principles and applications . 16; ; 15U Ae=o ^^-^ /- ^ Ans. loge V2 = Determine the following quantities (a) approximately by summation of a limited number of terms; (b) exactly by finding the limit of thesum of an infinite number of terms by integration. 3. The area under the curve y = a?, from x = 0 to x = 2; froma; «= —1 to X = 1. 4. The distance passed over by a body falling with constant accelera-tion g = per sec.^, from t =^ 1 to t = 4, v = gt being the relationof V and t. VOLUMES 267 6. The increase in speed of a body falling with acceleration
. Applied calculus; principles and applications . 16; ; 15U Ae=o ^^-^ /- ^ Ans. loge V2 = Determine the following quantities (a) approximately by summation of a limited number of terms; (b) exactly by finding the limit of thesum of an infinite number of terms by integration. 3. The area under the curve y = a?, from x = 0 to x = 2; froma; «= —1 to X = 1. 4. The distance passed over by a body falling with constant accelera-tion g = per sec.^, from t =^ 1 to t = 4, v = gt being the relationof V and t. VOLUMES 267 6. The increase in speed of a body falling with acceleration ofg = per , from t = Otot = 3. 6. The number of revolutions made in 5 minutes by a wheel whichrevolves with angular speed co = ^VlOOO radians per second. 7. The time required by the wheel of Ex. 6 to make the first tenrevolutions. 155. Volumes. — The volumes of most solids may befound approximately by the summation of a finite number ofparts and exactly by finding the limit of the sum of an infinitenumber of terms by Example. — To find the volimie of the right circular conewhose altitude is h and the radius of whose base is a. Divid-ing the volume into parts, each A 7, by passing planes Axapart parallel to the base Ah, and denoting a section at adistance x from the vertex at the origin by Ax, then, sinceAx/Ah = x^/h^, V is given approximately by XA7 = T AxAx = Xand exactly by Ah ^ Ax 7 = hm X ^k Ax=0 0 ^ Ah x^>/i2 3 h^ Ax ^Ah ph Jo x^dx I ,. (1) (2) (3) While AF is a frustum of the cone, dV may be representedby the cyhnder PMMi = Ax Ax = iry^dx. It is to be noted that the equations all apply to a pyramidwith any plane base Ah^^ well as to the cone. 268 INTEGRAL CALCULUS For another example: to find the volume of a sphere withradius a, divide by planes perpendicular to OX; then, since V = lim X A,^-^^x = ^ r {a - x^) dx Ax=0 —a Ci (X *J — a Aof . x^l Ao 4 „ 4 „ - . „ = -r a^a; — 17 = —? * o ^ = o 7^o^^ where Ao = wa^,c? \_ 3J_a
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