Name: ___________________________
Period _________
Honors Calculus
Date: ___________________
Mr. Montana
Intro to Integrals Review Sheet
1. Evaluate the following indefinite integrals.
a.
 (8 x
b.
 ( x  1)(3x  2) dx
c.
x
d.
 4x
e.
t
f.
t 2  2t  3
 t 4 dt
2
3
 9 x 2  4) dx
(2 x 2  1) dx
1
2
1
t
dx
dt
y 2  y  12
dy
( y  4)
g.

h.
 (sec
i.

j.
 1  cos
2
2
x  sin x)dx
 2 csc 2  d
cos x
2
x
dx
x
1
2
3
4
5
6
7
f(x)
4
8
16
20
24
32
40
2. The function f is continuous on the closed interval [1,7] and has values that are given in the table above.
7
Using the subintervals [1,3], [3,6], and [6,7] what is the trapezoidal approximation of
 f x dx ?
1
3. Oil is leaking from a tanker damaged at sea. The damage to the tanker is worsening as evidenced by the
increased leakage each hour, recorded in the following table.
8
Approximate  " leakage" dx with 4 left endpoint rectangles.
0
4. Find the general solution of the differential equation:
dy
 3x 2  4
dx
Find the particular solution if the original function passes through the point (2,6):
5. Find particular solution if f ‘(x) = 2 x 3  4 x and f (2) = 3.
3
6. Given

6
f ( x) dx  4 and
0
 f ( x) dx   1 , evaluate.
3
6

a.
3
f ( x) dx
b.
0
6
3
c.

6
f ( x) dx
3
6
e.
 f ( x) dx
0
 f ( x) dx
d.
  5 f ( x) dx
3
7. Suppose that f and g are continuous functions with the below given information, then use the properties
of definite integrals to evaluate each expression.
9
9
 f ( x)dx  1
1
 f ( x)dx  5
 g ( x)dx  4
7
7
1
(a)
 f ( x)dx
9
7
(b)
 f ( x)dx
1
7
(c)
  g ( x)  f ( x) dx
9
9
(d)
 2 f ( x)dx
1
9
(e)
  f ( x)  g ( x) dx
7
9
(f)
  2 f ( x)  3g ( x) dx
7
b
8. Write as a single integral in the form
 f ( x) dx .
a
3
a.

0

f ( x) dx +
4
0
b.
3
f ( x) dx -
 f ( x) dx
1
2
5
1
2
2
2
9
 f ( x)dx   f ( x)dx   f ( x)dx 
9. Evaluate the following Definite Integrals:
2
a.
 6 x dx
0
4
b.
3
x dx
1
7
c.
 6x
2
 2 x  3 dx
1

1  sin 2 x
0 cos 2 x dx
4
d.

e.
 (1  sin x) dx
0
1
f.
 (2t  1)
2
dt
0
 x  2 x2
1  x

9
g.

 dx

2
10. Approximate  1  x 2 dx using 3…
1
(a) Left endpoint rectangles
(b) Right endpoint rectangles
(b) Midpoint rectangles
(c) Trapezoids