Math 3A
4.7 – 4.9, 5.1 – 5.6, 5.8, 6.1 – 6.2, 6.3.
Final Review I
The problems that follow are intended as extra practice for the exam. You are responsible for all material
covered in class, on homework and on quizzes. There may be types of problems on the exam that are not
shown here.
1. A particle moving along a straight line is accelerating at a constant rate of 5 m/sec/sec. Find the
initial velocity if the particle moves 60 meters in the first 4 seconds.
2. A car traveling 75 miles/hour along a straight road decelerates at a constant rate of 12 ft / s 2
Note: 88 ft / s  60mi / hr
a. How long will it take until the speed is 45 mph?
b. How far will the car travel before coming to a stop?
3. Evaluate each definite integral
a.

1

b.
2
2
2  x dx
c.

5 x  1dx
1
9

 sec  3x  dx

2
d.
12
4
 tan
2
 sec2  d
0
4. Evaluate each indefinite integral
sin 5 x
a. 
dx
1  cos 5 x
b.  cot x dx
c.
dx
 x ln x
5. Evaluate the definite integral by expressing it in terms of u and evaluating the resulting integral


using a formula from geometry.
e3
6. Evaluate the integral.
7.
e
3
9   ln x 
x
2
sin  1  4cos 2  d
3
2
dx
  
Find the average value the function f  x   sec2 x over the interval   , 
 4 4
8. Find the force on a 100 ft wide by 5 ft deep wall of a swimming pool filled with water. Neglect the
effect of the atmosphere above the liquid. The density of water is 62.4 lb/ft 3 .
9.
a. A spring exerts a force of 0.5 N when stretched 0.25 m beyond its natural length. Assuming
that Hooke’s law applies, how much work was performed in stretching the spring to this length?
b. How far beyond its natural length can the spring be stretched with 25 Joules of work?
1
4 ft
4 ft
10. A flat surface is submerged vertically in water as shown.
Find the fluid force against the surface. Assume that the weighs density of
water is 62.4 lb/ft 3
12 ft
10 ft
11. The rectangular tank shown here, with its top at ground level, is used to
catch runoff water. Assume that the weighs density of water is 62.4
lb/ft 3
ground level
a. How much work does it take to empty the tank by pumping the water back
to ground level once the tank is full?
b. If the water is pumped to ground level with a (5/11)-horse-power motor
(work output 250 lb  ft / sec ), how long will it take to empty the full tank
(to the nearest minute)?
20
c. Show that the pump in part (b) will lower the water level 10 feet (halfway) during the first 25
minutes.
12. Find the area enclosed by the curves
1
a. y  x  2 and y  x  2
2
2
b. y  2  x and y  x
c. y  x  1 and y 2  2 x  6
d. x  y 2  y and x  y3  4 y 2  3 y
13. Use the method of disks/washers to find the volume of the solid that results when the region
enclosed by:
a. y  4 x  16 , y  0 , and x  1 is revolved about the x-axis.
b. x  y 2 , x   y  6 is revolved about the y-axis.
c. x  y 2 and x  y is revolved about the line x  1 .
14. Use cylindrical shells to find the volume of the solid when the region enclosed by:
a. y  x 3 , x  1, x  2 and y  0 is revolved about the y-axis.
b. y  x3 , y  1 and x  0 is revolved about the line y  1 .
15. Find the area of the surface that is generated by revolving the portion of the curve y  x3 between
x  0 and x  1 about the x-axis.
16. Find the area of the surface that is generated by revolving the portion of the curve x 
y  1 and y  4 about the y-axis.
y between
2
17. Assume that y  f  x  is a smooth curve on the interval  a, b  and that f  x   0 for a  x  b .
Describe a formula for the surface area generated when the curve y  f  x  , a  x  b , is revolved
about the line y  k
 k  0 .
18. Let R be the region in the first quadrant enclosed y  x 2 , y  2  x and x  0 . In each part, set
up, but do not evaluate, an integral or a sum of integrals that will solve the problem.
a. Find the area of R by integrating with respect to x
b. Find the area of R by integrating with respect to y
c. Use the method of disks/washers to find the volume of the solid generated by revolving R about
the x – axis.
d. Use the method of disks/washers to find the volume of the solid generated by revolving R about
the y – axis.
e. Use cylindrical shells to find the volume of the solid generated by revolving R about the y –
axis.
f. Use the method of disks/washers to find the volume of the solid generated by revolving R about
the line y  3 .
g. Use cylindrical shells to find the volume of the solid generated by revolving R about the line
x  2 .
x6  8
from x  2 to x  3
16 x 2
1
1
20. Consider the curve segments y  x 2 from x  to x  2 and y  x from x  to x  4
2
4
a. Graph the 2 curves segments and use your graphs to explain why the lengths of these 2 curves
segments should be equal.
b. Set up integrals that give the arc lengths of the curve segments by integrating with respect to x.
Demonstrate a substitution that verifies that these 2 integrals are equal.
19. Find the exact length of the curve y 
21. Describe the method of slicing for finding volumes and use the method to derive an integral
formula for finding volumes by the method of disks.
22. Find
a.
dy
. Use algebraic properties of the natural logarithm function when helpful.
dx
y  ln  tan x 
b. y  x ln  2 x 
c.
y   ln x 
3
 x 3 x 1 
d. y  ln 

 sin x sec x 
3
23. Find
a.
b.
24. Find
a.
dy
use logarithmic differentiation
dx
sin x cos x tan 3 x
y
x
y  xsin x
dy
dx
y  log5  x 2  7 x  5


b.
y  x log 2 x 2  2 x 
c.
ye
3
15x3
25. Find the equation of the tangent line to the curve
y  ln x at x  e1
26. Find the equation of the tangent line to the graph of y  ln x at x  2
27.
a. Find the equation of the line through the origin that is tangent to the graph y  ln x
b. Explain why the y-intercept of a tangent line to the curve y  ln x must be 1 unit less than the ycoordinate of the point of tangency.
28. Let f  x   x3  2 e x
a. Show f is a 1-1 function
b. Find f 1  2
  
4