Question

Find the volumes of the solids of revolution when the region bounded by the curves y = f(x) and y = g(x) is rotated about the x-axis. (Remember that to find intersection points involving transcendental functions such as In (x), it is necessary to use the FindRoot command in place of the Solve command, as in FindRoot [f[x] ==g[X], {x,a}], where a is a first approximation to the root in question. A good estimate to a may be obtained by first graphing f(x) and g(x)). 1. f(x) = 2 x3 – 3x+2, g(x) = 2 x2 +1. 2. f(x) = In (x)+4, g(x) = e' + e*. 3. f(x) = 3 cos(x+2 ), g(x) = -(x-3)2.

Answer 1

Solm we Volume obtain by Rotating region flor) -g2 (x)} da a b point of intersection in f(x), g(x) Graph and solution shown below (a) 1(x) = 2x3 – 3x+2 g(x) = 2x2 +1 (-1,3) 0.293 (0.293, 1.172) л * (82(x)2 – 8ı (x)2) dx - 10 -5 0 5 = 20.2691161111 Volume = 20-26912 (6) Graph and solutim for volume Volume=19.69 371 (1.411, 4.344) w $2(x) = ln x + 4 (0.138, 2.019) 82(x) = ex +e-* -10 -5 0 5 1.411 TT = 19.6937100962 (C) Graph and volume of region of revolution shown below. $3(x) = 3 cos(x + 2n) 4 g,(x) = -(x - 3)^ (1.948, -1.106) (4.206, -1.455) 4.206 TT °((f3(x))2 – (83(x))2)dx 1.948 = 40.3464989995 10 -5 11 Volume=40.3465 12

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