Unformatted text preview: 3B Practice Problems for Final Spring 2006 The ﬁnal is on Tuesday, June 13, 8:00 – 11:00am at our class room. You may
bring a 3 by 5 card.
I will have extra oﬃce hour on 6/9 Friday, 6/12 Monday 9-11am at SH 6503.
The ﬁnal will be from sections 5.1-5.5, 6.1-6.5, 7.1-7.8, 8.1-8.3.
Subjects you need to know well and use well:
The fundamental theorem of calculus, basic integration formulas, substitution
rule, integration by parts, integration of trig functions, radical functions, rational
functions, improper integral.
Application of integration to distance, area, volume, work, arc length, surface area.
A good starting point for preparing the ﬁnal is doing the problems from previous
practice and midterms.
Following are some more practice problems.
1. Find the derivative of the function y = cos x cos(u2 )du.
2. Find the following integrals.
(a) es cos(es )ds (b)√01 (e − ex )x dx
(d) 02 t 1 + √ dt
(c) lnx x dx
(e) sin(3x)e2x dx
(f) 12 xx−1 dx
(g) sin2 x cos3 xdx
(h) x2 +4x+8 dx
(i) x2 +5x−6 dx
(j) x2 −61x+10 dx
(k) 0 x ln xdx
(l) Is 1+∞ √x2 +3+1+7x7 dx convergent or divergent?
3. A 1000-lb cube of ice must be lifted 50 ft, and it is melting at a rate of 2 lb
per minute. Assume that it can be lifted at a rate of one foot every minute. Find the
work needed to get the block of ice to the desired height.
4. a) Find the length of the curve
x y= √ t − 1dt 1 ≤ x ≤ 16 1 b) Find the area of the surface obtained by rotating this curve about y-axis.
5. Determine the volume of the solid obtained by rotating the region bounded by
the function f (x) = ex/3 , the lines x = 0, x = 1 and the x-axis around the line y = 4.
6.Use the Trapezoidal Rule and the Midpoint Rule to approximate the integral
sin(x2 )dx with n = 4.
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- Fall '07