c
Math 151 WIR, Spring 2013, Benjamin
Aurispa
Math 151 Week in Review 14
Sections 6.1-6.4
1. Calculate the following sums.
(a)
5
X
(2i2 + 3)
i=1
(b)
6
X
(−1)i (i − 5)2
i
i=3
(c)
35
X
i=5
(d)
(e)
20
X
i=1
n
X
3i − 3i+1
i3 +
25
X
3
i=4
(3i + 2)2
i=1
2. Represent the following sum using sigma notation.
3. Calculate lim
n→∞
" n
X
4i 2
4
i=1
n
n
4 9 16 25
400
+ +
+
+ ··· +
7 8
9
10
25
#
4i
+7
+
n
4. Approximate the area under the graph of f (x) = x2 + 1 on the interval [0, 8] by using the partition
P = {0, 2, 3, 6, 8} and taking x∗i is taken to be the left endpoint.
5. Approximate the area under the graph of f (x) = ln x on the interval [1, 11] using 5 equal subintervals
and taking x∗i to be the midpoint of the subinterval.
6. Approximate the area under the graph of f (x) = cos x + 2 on the interval [0, 3π
2 ] by using 6 equal
∗
subintervals and taking xi to be the right endpoint.
7. Set up a limit to find the exact area under the curve f (x) = 36 − x2 between x = 2 and x = 5.
8. Evaluate the following definite integrals by interpreting in terms of areas.
(a)
Z
3
Z
8
Z
5
−3
(b)
p
9 − x2 dx
f (x) dx where f (x) =
0
(c)
|2x − 8| dx
0
(d)
Z


 3
if 0 ≤ x ≤ 2
if 2 < x < 6
if 6 ≤ x ≤ 8
5−x

 −1
4
−3
(|x| − 3) dx
9. Write the folowing as a single integral.
Z
2
7
f (x) dx −
1
Z
4
7
f (x) dx +
Z
2
−1
f (x) dx
c
Math 151 WIR, Spring 2013, Benjamin
Aurispa
10. Calculate the derivatives of the following functions.
(a) h(x) =
Z
x
t2 + 1 dt
−4
(b) g(u) =
Z
u3
π
(c) h(x) =
Z
p
1
dt
1 + t4
7
sin s ds
tan x
11. Evaluate the following integrals.
(a)
Z
2
1
(b)
Z
1
(1 − 2x − 3x2 ) dx
√
√
(u + 1)( u + 3 u) du
0
(c)
Z
1
(d)
Z
(e)
Z
5 2
t
t
+e
dt
−2
|9 − x2 | dx
−4
π/4
−π/2
(cos u − sin u) du
12. Find the general indefinite integral
Z 2x − sec2 x +
7
1 + x2
dx.
13. A particle has velocity function given by v(t) = t2 − 4t + 3.
(a) Find the displacement of the object during the first 5 seconds.
(b) Set up an expression involving integrals to find the total distance traveled by the object during
the first 5 seconds.
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