7.5. VIDEOS 189

Note how this does an approximation for 0,h,2h, then from 2h,3h,4h, etc. Denotethis as S (a,b, f ,2m) denote the approximation to

∫ ba f (x) dx obtained from Simp-

son’s rule using 2m+1 equally spaced points with x0 at the left. Show∣∣∣∣∫ b

af (x) dx−S (a,b, f ,2m)

∣∣∣∣< M1920

(b−a)5 1m4

where M ≥ max{∣∣∣ f (4) (t)∣∣∣ : t ∈ [a,b]

}. Better estimates are available in numerical

analysis books but these also have the error in the form C(1/m4

).

27. Suppose fn converges uniformly to f on [a,b] and that fn ∈ R([a,b]) . Show f ∈R([a,b]) and that

∫ ba f (x)dx = limn→∞

∫ ba fn (x)dx. That is, the uniform limit of Rie-

mann integrable functions is Riemann integrable.

28. Find a power series to approximate ln(1− x) about 0 and show the remainder termconverges to 0 if |x|< 1.

29. Find a power series to approximate ln(1+ x) about 0 and show the remainder termconverges to 0 if |x|< 1.

30. Give a series which will approximate ln( 1+x

1−x

)whenever |x| < 1. Show that for any

r > 0, there is x, |x|< 1 such that 1+x1−x = r. Explain why the partial sums of the series

will converge to lnr.

31. A two dimensional shape S in a plane has area A and a cone is formed from drawingall lines from a point in S to a single point at height h above S. By similar triangles,the linear dimensions of similar shapes at height h in a plane parallel to the givenplane are h−y

h times the corresponding ones in the plane at the base. Thus if A(y) isthe area of the cross section in the plane at height y corresponding to S, it followsthat the cross section at height y has area A(y) = A

h2 (h− y)2. Find the volume of the

cone. Hint: Show∫(h− y)2 dy =− (h−y)3

3 +C and use this. This includes pyramids,tetrahedra, circular cones, etc.

7.5 VideosRiemann integral