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Final Exam Review - Integral Calculus | M 408L, Exams of Mathematics

Material Type: Exam; Class: INTEGRAL CALCULUS; Subject: Mathematics; University: University of Texas - Austin; Term: Fall 2008;

Typology: Exams

Pre 2010

Uploaded on 08/31/2009

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M408L Final Exam Review – Day 1
1) A car is traveling 60 mi/hr when the brakes are fully applied, skidding the car to a stop
at a constant deceleration of
2
22 ft /sec
. What is the length of the skid mark on the
highway? (Hint: 60 mi/hr = 88 ft/sec)
2)
A) Write a Riemann sum expression (using
n
subintervals) for the area of the
region under the graph of
4
( )f x x
on the interval
[0,3]
B) Determine the definite integral
I
to which the Riemann sum
3
1
4 4
2
n
i
i
n n
corresponds.
i)
23
0
(2 )I x dx
ii)
63
2
I x dx
iii)
iv)
53
1
(2 )I x dx
pf3
pf4
pf5

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M408L Final Exam Review – Day 1

  1. A car is traveling 60 mi/hr when the brakes are fully applied, skidding the car to a stop

at a constant deceleration of

2

22 ft / sec. What is the length of the skid mark on the

highway? (Hint: 60 mi/hr = 88 ft/sec)

A) Write a Riemann sum expression (using

n subintervals) for the area of the

region under the graph of

4

f ( ) xx

on the interval [0,3]

B) Determine the definite integral I to which the Riemann sum

3

1

n

i

i

n n

corresponds.

i)

2 3

0

I  (2  x ) dx

ii)

6 3

2

Ix dx

iii)

3

4

0

x

I dx

iv)

5 3

1

I  (2  x ) dx

A) If

2

2cos( )

( ) ln( )

t

x

F x t e dt

 

Find

/

F (0)

B) For

3

2

2

x

t

x

G x  e dt

Find G '( ) x

  1. Evaluate the integral

4

0

4sin 6

sin 3

x

I dx

x

  1. Determine the indefinite integrals

A)

2

3 3

x

dx

x x

B)

3

x

dx

x

C)

3 4 2ln

6

x x

e dx

  1. Let  be the area bound between the functions:

2

( )

x

f xe , y^ ^4 , x  0

A) Write an integral which expresses the volume of the solid formed by rotating

the area  about the x-axis.

B) Write an integral which expresses the volume of the solid formed by rotating

the area  about the y-axis. What technique is necessary to evaluate such an

integral?

  1. Determine the indefinite integrals

A)

2

2

x x

dx

x x

B)

2

x

dx

x

C)

dx

xx

  1. Determine the limits

A)

6 0

lim

x

x

e

x x

B)

0

3 ln

lim

x

x x

x

 

C) (^)  

5

0

lim cos3 x

x

x

D)

1

lim

ln 1

x x x

  1. Determine the integrals

A) x^ cos(5 ) x dx

B)

3

x ln( ) x dx

C)

2

sin(3 )

x

x e dx

D) sin^ xdx

  1. Determine the integrals

A)

2

tan x dx

B)

3

4

sin

cos

x

dx

x

C)

2 2 3

(cos x sin x ) dx

  1. Determine the indefinite integral

A)

5

5

2 2

2

1

25

dx

xx

B)

2 2 3/ 2

1

(( ) )

dx

axb

C)

0

2 2 1 ( 2 2)

dx

x x

  

D)

3

2

2

2 5/ 2 2 (5 4 )

dx

x x

 

  

 

  