MATH 101 HOMEWORK 7
Due on Wednesday, November 5
Covers sections 6.7, 6.8, 7.1. For full credit, show all work.
2T2n + Mn
, where S2n , Tn , Mn denote Simpson’s, trapezoid, and
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Rb
midpoint approximations as usual to the integral a f (x)dx.
1. (5) Verify that S2n =
Z
2. (10 marks) We want to evaluate numerically the integral
∞
p
x−3 + x−6 dx. For this
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purpose, we want to use a substitution of the form x = t−a to convert the integral to one
that can be handled by numerical methods.
(a) For what values of a does the integral become proper (i.e. finite interval of integration
and no infinite limits)?
(b) In order to have good error estimates for the Midpoint Rule approximation, f 00 (x)
should be bounded on the interval of integration (i.e. |f 00 (x)| ≤ K for some constant K).
Find a value of a for which this happens. (The problem can be solved without a graphing
calculator, but it could also be interesting to use one to graph the function for several
different values of a.)
3. (5 marks) Set up (but do not evaluate) the integral for the volume of the solid obtained
by rotating the finite region bounded by y = 2x2 and y = x2 + 1 around the y axis. You
may use either slices or cylindrical shells, whichever you prefer.
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