1
Math 190: Calculus I
Test 4: Integrals
(1) Show your work clearly, do not just give your answer.
(2) Try to give some explanation to what you are planning to do.
Name: __________________________
Number
1
2
3
4
5
6
7
8
9
10
Points Your score Comment
15
10
7
16
6
7
15
15
18
11
Total
120
Your Test 4 score = _____________________
Your approximated course grade = ______________________
2
1. (a) Estimate the area under the graph of f ( x)  1  x from x = -1 to x = 2 using
six rectangles and right endpoints.
2
(b) Repeat (a) using midpoints.
© Which is a better estimate?
3
2. Express the limit as a definite integral on the given interval.
ex
x , [1, 5]
(a) lim 
n 
i 1 1  xi
n
i
n
(b) lim
n 
[4  3( x
i 1
*
i
) 2  6( xi ) 5 ]x ,
*
[0, 2]
3. Express the integral as a limit of Riemann sum. Do not evaluate the integral.
10

1
( x  4 ln x) dx
4
4. Use the Fundamental Theorem of Calculus to find the derivative of the function.
u
(a) g (u )  
3
(b) y 
x

3
1
dx
x  x2
cost
dt
t
5
5. Find the general indefinite integral.
64
6. Evaluate

1
1 3 x
dx .
x
 1  t  2  t dt .
2
6
7. The boundaries of the region are the y-axis, the line y = 2 and the curve y 
Find the area of this region. (First sketch the region.)
8. The velocity function(in m/s) is given for a particle moving along a line.
v(t )  t 2  2t  8
(a) Find the displacement and
(b) the distance traveled by the particle during 1 ≤ t ≤ 6.
4
x.
7
9. Evaluate the integral by substitution.
(a)
 x
(b)

x
2
 1
sin
2
dx
 x  dx
x
8
©

x 1  x 3 / 2  dx
 x 
1
10. Evaluate
0

1  x 2 dx by interpret it in terms of areas.