M216
TEST #2
Name:
Show ALL your WORK! For each problem, qualify your answer. See table below for points.
ANSWER (A) certain (B) fairly sure (C) not sure
EXTRA POINTS for each problem:
Correct
+1
+ 12
− 12
Wrong
−1
− 12
+ 12
1. Find the area of the region enclosed by the curves  =  − 1 and the parabola  2 = 2 + 6
 2 = 2 ( + 1) + 6 Solution is : 4 −2
¶
µ
¶=4
µ
µ 2
¶¶
Z 4µ
1
1
 −6
1
=
= 18
( + 1) −
−  2 +  + 4  = −  3 +  2 + 4
 =
2
2
6
2
−2
−2
=−2
Z
4
¤ How sure are you about your answer to the problem above?
(A) I am certain I am right!
(B) Fairly sure
(C) Not sure at all
2. Find the area of the region enclosed by the curves  = || and  = 2 − 2 
y
2.5
2.0
1.5
1.0
0.5
-2
 =
Z
1
−1
-1
1
-0.5
2
x
|| = 2 − 2 : solution is  = −1 and  = 1
µZ 1
¶
¡¡
¢
¢
¡
¢
2
2
2 −  − ||  = 2
2 −  −  
0
µ
µ
¶=1
¶
µ
¶
12 − 2 − 3
1
1
1 1
7
= 2 2 − 3 − 2
=2 2− −
=2
=
3
2
3 2
6
3
=0
¤ How sure are you about your answer to the problem above?
(A) I am certain I am right!
(B) Fairly sure
(C) Not sure at all
3. Find the volume of the solid that results when the shaded region, bounded by the curves  = 2 − 2   = , and
 −  is revolved around the  −  :
y
2
2
1
1
-2
0
0.2
0.4
0.6
0.8
-1 0.00
0
z
x
y
0.21
0.4
2
1.0
0.8
0.6
1.0
x
-1
-1
-2
-2
Z
1

0
µ
¶=1
¡
¢
1 5 5 3
 4 − 52 + 4  = 
 −  + 4
5
3
0
=0
µ
¶
1 5
38
= 
− +4 =
 ≈ 796
5 3
15
Z
³¡
´
¢2
2 − 2 − 2  =
1
¤ How sure are you about your answer to the problem above?
(A) I am certain I am right!
(B) Fairly sure
(C) Not sure at all
4. Find the volume of the solid that results when the region enclosed by  =  and  = 2 is revolved about the
 −  (Use any method you want.)
y
1
1
x
0.0
0.5
0 0.3 0.4
0.1 0.2
1
2
0.6
0.7
0.8
0.9
1.0
x
-1
µ
¶=1
¶
1 1
1 3 1 5
2
=
 − 
−
=

3
5
3
5
15
0
0
=0
!=1
à 5
µ
¶
Z 1³
Z 1
´
3
2
1 3
2 1
2
√
2
2
 −   = 2 5 − 
2 () (  − )  = 2
= 2
 :  =
−
=

3
5
3
15
0
0
2
  :  =
Z
1
Z
³
¡ ¢2 ´
 ()2 − 2
 =
1
¡
¢
 2 − 4  = 
µ
=0
¤ How sure are you about your answer to the problem above?
(A) I am certain I am right!
(B) Fairly sure
(C) Not sure at all
5. Find the volume of the solid whose base is the region bounded between the curves  = 1, 3 = 4 −  and  = 4
and whose cross sections perpendicular to the − axis are squares. (Note: the shaded triangle below does NOT
touch the -axis!)
Z
4
1
µ
¶2
¶
µ
¶=4
Z 4µ
1
1 2 2
1
1 3 1 2 1
=  = 1
1 − (4 − )  =
 − +
 =
 −  + 
3
9
9
9
27
9
9 =1
1
or
1
1
 ·  = (  = 1 × 1 pyramid of height  = 3) = (1 · 1) (3)
 =
3
3
¤ How sure are you about your answer to the problem above?
(A) I am certain I am right!
(B) Fairly sure
(C) Not sure at all
1
6. Find the volume of the solid generated when the region enclosed by  =   = 0  = 1 and  = 3 is revolved

around the  − 
 =
Z
3
2 ()
1
µ ¶
1
 = 4

¤ How sure are you about your answer to the problem above?
(A) I am certain I am right!
(B) Fairly sure
(C) Not sure at all
7. Find the volume of the solid generated when the region bounded by the graphs of  =   = 0 and  = 1
revolves around the line  = −1
 =
Z
1
0
or
Z 1 ³
´
4
4
2
2 (1 + ) (1 − )  =  or  =
 (1 + ) − 12  = 
3
3
0
¤ How sure are you about your answer to the problem above?
(A) I am certain I am right!
(B) Fairly sure
(C) Not sure at all
8. Find the average value of the function  () = 1 + 2 on the interval [−1 2]!
¶=2
µ
µ
µ
¶¶
Z 2
¡
¢
1
1
1
1 3
8
1
2
 =
1 +   =
=
+ 
2 + − −1 −
=2
2 − (−1) −1
3
3
3
3
3
=−1
¤ How sure are you about your answer to the problem above?
(A) I am certain I am right!
(B) Fairly sure
(C) Not sure at all