Student (Print)
Section
Last, First Middle
Student (Sign)
Student ID
Instructor
MATH 152
Exam1
Spring 2000
Test Form A
1-10
/50
11
/10
12
/10
Part II is work out. Show all your work. Partial credit will be given.
13
/10
You may not use a calculator.
14
/10
15
/10
Part I is multiple choice. There is no partial credit.
TOTAL
1
1-10
/50
11
/10
12
/10
13
/10
14
/10
15
/10
TOTAL
Formulas:
sin A sin B 1 cos A " B " 1 cos A B
2
2
sin A cos B 1 sin A " B 1 sin A B
2
2
1
1
cos A cos B cos A " B cos A B
2
2
1
f x 0 4f x 1 2f x 2 4f x 3 2f x 4 2f x n"2 4f x n"1 f x n
Sn 3
3
K b"a
where K u |f x | for a t x t b
|E T | t
12n 2
K b"a 5
where K u f Ÿ4 x for a t x t b
|E S | t
180n 4
; sec 2 d2 ln|sec 2 tan 2 | C
Ÿ Ÿ Ÿ Ÿ Ÿ
Ÿ
Ÿ
Ÿ
Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ C Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ x
UU
; csc 2 d2 ln|csc 2 " cot 2 | C
; ln x dx x lnx " x C
2
Part I: Multiple Choice (5 points each)
There is no partial credit. You may not use a calculator.
1. Compute ;
1/2
xe 2x dx.
0
a. 1 e " 1
b.
c.
d.
e.
2
4
1 e2
4
1 e2 " 1
4
1
4
3
4
Ÿ
=
Ÿ 2. Compute ; cos 2 x dx.
0
a. 1
b. =
2
c. 2
d. = " 1
2
e. =
3. Find the average value of the function y xe x on the interval 0 t x t 2.
2
Ÿ
Ÿ
a. 1 e 4 " 1
b.
c.
d.
e.
2
1 e4 " 1
4
e
1 e4
2
e4
3
5
4. Use the Trapezoid Rule with n 2 to approximate ; 1
x dx.
1
a. 14
15
b. 23
15
c. 28
15
d. 46
15
e. 76
45
5
5. If you use the Trapezoid Rule with n 8 to estimate ; 1
x dx you obtain the approximation
1
T 8 1.628968. (Don’t compute this.) Use the Trapezoid Error formula
K b"a 3
where K u |f x | for a t x t b
|E T | t
12n 2
to find an upper bound for the error in this approximation.
a. |E T | t 1
6
b. |E T | t 1
60
c. |E T | t 1
72
d. |E T | t 1
300
e. |E T | t 1
750
Ÿ
6. Compute ;
a.
=/4
0
UU
Ÿ Ÿ Ÿ tan 2 x sec 4 x dx
2
2
" ln 1 "
2
2
2
ln 1 " 2
2
2
c. 8
15
d. 4
35
e. 4
b.
4
7.
Find the volume of the solid obtained
by rotating the region bounded by
y x 3 " x and y 0 for x u 0
about the y-axis.
a.
b.
c.
d.
e.
4 =
15
1=
2
2 =
15
8 =
105
16 =
105
2
8.
Which integral gives the volume
of the solid obtained by rotating
the region bounded by
x 2 y 2 2 and y 1
1
0
-1
about the x-axis?
-2
a. ;
b. ;
1
"1
Ÿ2 " x 2 2
=
2
" 2
=
2 " x2 " 1
2
dx
Ÿ
1
" 1 dx
c. ; 2=x 1 " x 2 dx
0
1
d. ; 2=x
e. ;
0
1
"1
Ÿ
2 " x 2 " 1 dx
= 1 " x 2 dx
5
9.
Find the area of the region
bounded by
x 3y 2 ,
x " 2y "2,
y "2 and y 3.
a.
b.
c.
d.
e.
8
24
40
56
64
10. Which of the following definite integrals equals ;
a. 4 ;
b. 2 ;
c. 2 ;
d. 4 ;
e. 4 ;
=/2
0
=/2
0
=/4
0
=/4
0
=/4
0
3
1
x 2 " 2x 5 dx?
sin 2 d2
sin 2 2 d2
tan 2 d2
sec 2 d2
sec 3 2 d2
6
Part II: Work Out (10 points each)
Show all your work. Partial credit will be given.
You may not use a calculator.
11.
An object has a base that is
2
the triangle with vertices
Ÿ0, 0 , Ÿ1, 0 ,
and
Ÿ0, 2 .
The crosssections perpendicular
y
to the x-axis are semicircles.
Find the volume of the object.
You must use an integral.
12. Compute ;
x
2
0
x
1
1
dx.
x2 " 4
7
13.
A tank has the shape of a circular cone with height 20 m,
radius 8 m and vertex down. It has a spout which extends
5 m above the top of the tank. The tank is initially full of
water. Set up the integral that gives the work required to
pump all the water out of the spout. Do not evaluate the
integral.
water density 1000
kg
m3
g 9.8 m 2
sec
e
14. Compute ; x lnx dx.
1
8
15.
a. Find the partial fraction expansion for x3 " 2 .
x x
b. Then compute ; x3 " 2 dx.
x x
9