100
Math 171-01:Quiz #1 (5.1, 5.3, 5.4)
Name:
Show as much work as possible to get full credit.

1. Use the graph of y  f  x  shown below to find the exact value of the definite integrals.
(a)
 f xdx
6
0

(b)
(4 points each)
= 4(5) – ½(2)(4+2)
= 20 - 6
= 14
 f xdx
12
4
= -½(4)(4) + ½(π)(22)
= 2π - 8

(c)
 f xdx
12
5
= -½(3)(3) + ½(π)(22)
= 2π – 4.5

2. Find each of the following definite integrals by using the geometric definition of integrals. That is, do
not use the Fundamental Theorem of Calculus. Show some work. (6 points each)
(a)

6
x  2 dx
1
½(1)(1) + ½(4)(4) = 8.5

(b)

4
 16  x 2 dx
4
-½(π)(42) = -8π

(c)

2
0
sin xdx
= 0 because the “areas” cancel each other out

3. Suppose we know that
 f xdx  8,  f xdx  2, and  f xdx  3.
4
4
6
0
2
4
(6 points each)

 3

 3 f x 5dx
2
0



 f xdx   f xdx  
4
4
2
0
2
0
 38  2 
52  0
Find
 3f x 5dx .
2
0
5dx

18 10
8
4. Consider
the function Fx

 f tdt where f is the function shown below.
x
(5 points each)
0
(a) Over what intervals in [0, 10] is
F decreasing?

[5, 8]
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
(b) Over what intervals in [0, 10] is
F concave up?
[1, 3]  [6, 10]
5. Find the exact value of the following using the Fundamental Theorem of Calculus.
(a)
 x
3
1


2

 4 x  2 dx
x
1
3
3
 2x 2  2x
(b)




cos2x dx
4
6

 12 sin 2x 
3
1
6

  103
 12 1 12
1
3


4
  18  6   2 
2
27
3
(7 points each)
  12 sin3 
1
sin 2
2
2 3
4

3
2
6. Find each of the following integrals. Give exact answers for definite integrals.
(a)

e
sinln x
x
1


1
0
(8 points each)
Let u  ln x  du  1x dx
dx
1
sin udu  cos u 
 cos1 cos0


0
1 cos1

(b)
x4
 1 x
2
dx
 1 x
x

2
dx 

 12

1
u
 1 x
4
Let u 1 x 2
dx
2
du  4 arctanx  C


 12 ln 1 x 2  4 arctanx  C

(c)
 csc
2
x cot x dx Let u  cot x  du  csc2 xdx



(d)


3
  23 cot x  2  C
3
x
dx
x4
u4

du
u 


  23 u 2  C
udu

Let u  x  4  du  dx and x  u  4

 u
1
2
 4u
 12
du
 23 u 2  8u 2  C
3

2
3
1
x  4
3
2
8 x  4 C

(e)
 x  3sin x
3
 12

 u du
3
2
 

 6x cos x 2  6x dx
 18 u 4  C


  C
 18 sin 4 x 2  6x




Let u  sin x 2  6x  12 du  x  3cos x 2  6x
