Calculus Spring Exam Review
Formulas
n(n +1)
åi = 2
i =1
n
n(n +1)(2n +1)
åi =
6
i =1
n
å c = cn
i =1
n
2
n 2 (n +1)2
åi = 4
i =1
n
3
d u
du
éë a ùû = ( ln a ) au
dx
dx
d
1 du
log a u ] =
[
dx
( ln a ) u dx
ò tanu du = -ln cosu + C
ò cot u du = ln sinu + C
ò secu du = ln secu + tanu + C
ò cscu du = -ln cscu + cot u + C
òa
x
1 ö x
dx = æç
a +C
è ln a ÷ø
Chapter 4
1. Use the properties of sigma notation and the above summation formulas to evaluate
10
å (i
the sum:
2
+ 3i - 2).
i=1
2. (SKIP) Write the definite integral that represents the shaded region.
5
3. Draw a sketch representing
ò (-2x + 6)dx.
-1
4. If
6
12
12
-1
6
6
ò f (x)dx = 5 , ò f (x)dx = -3, and ò f (x)dx = 6,
find each using properties of the
definite integral.
12
a) Find
ò
12
f (x)dx
b) Find
-1
ò [ f (x) + 3]dx
6
5. Find the average value of f ( x)  3x  2 on the interval [0, 2].
2
6. Find f(x) if f " ( x)  x  2 , f ' (0)  3 , f (0)  1
7. Use a(t )  32 feet per second squared as the acceleration due to gravity. A ball is
thrown vertically upward from the ground with an initial velocity of 56 feet per second.
For how many seconds will the ball be going upward?
3
8. Use the Trapezoidal Rule with n = 4 to approximate
1
ò ( x - 1) dx.
2
2
9. Evaluate the following integrals by hand (not with a calculator). Give an EXACT
value (decimal approximations are not acceptable) for definite integrals.
a.

b.
ò 3csc
c.
x4  x3
 x 2 dx
d.
ò (3x
e.
sec 3 q tanq
ò 1+ tan2 q dq
f.
ò 2 cos ( 3x )dx
g.
ò 3x
x 2 dx
5
2
xdx
+ 1)2 dx
2
4x 2 -1dx
h. (SKIP) ò 3x 4x - 1dx
i.
ò sin(3x + 4)dx
2
j.
1
x
2
dx
1
p
k.
3
ò csc x cot xdx
p
4
x2 + 5
ò x 2 dx
-1
1
l.
m.
ò x(x
n.
ò sin
3
2
- 1)4 dx
3x cos 3xdx
1
o.
ò x dx
-1
p. SKIP ò x x + 1dx
Chapter 5
10. Find y.
a. f (x) = ln
b. y = x e
x 2 4x + 1
(x
3
+ 5)
3
x
c. f ( x) 
e2x
2x
d. f ( x)  (3 sin x  5) 4
e. f (x) = x ln x
f. f (x) = 4x 2 e3x
g. j(x) = cos3(2x)
h. ln xy  x  y
i. xy 3 - 3x = 2y - 5
j. y = 3xx3
11. Evaluate.
4e
a.
1
ò x dx
e
e+1
b.
1
ò x - 1dx
2
2x + 1
dx
x +1
c.
ò
d.
ò x cot x dx
e.
3x 2 + 3x + 3
ò x 2 +1 dx
2
x
f.
ò
e
g.
ò
ln x
dx
x
h.
cos 3 x - sin 2 x
ò cos2 x dx
i.
ò tan
x
5
dx
x sec 2 xdx
12. Find the general solution to the first order differential equation: (4-x)dy + 2ydx = 0.
13. A certain type of bacteria increases continuously at a rate proportional to the number
present. If there are 500 present at a given time and 1000 present 2 hours later, how
many will there be 5 hours from the initial time given?
Chapter 7
14. Determine the area of the region bounded by the graphs of y = -x2 + 2x + 3 and y = 3.
15. Find the area of the two regions bounded by y = 3 – x2, y = 3 – x, and x = -2.
16. Find the volume of the solid formed by revolving the region bounded by the graphs
of y = x3, x = 2, and y = 1 about the y-axis.
17. What integral represents the volume of the solid formed by revolving the region
bounded by the graphs of y = x3, y = 1, and x = 2 about the line y = 10?
18. Find the volume of the solid formed by revolving the region bounded by the graphs
1
of y = (x - 2)2 and y = 2 about the y-axis.
2
19. Find the volume of the solid formed by revolving the shaded region enclosed by
y = x , y = 4x , and x = 4 about the line y = 4.
20. Let R be the region bounded by f(x) = x2 and y = 2.
a. Sketch the region R.
b. Find the area of R.
c. Revolve R around the x-axis and find the volume of the solid generated.
Cross sections