Exam I and Key
July 12, 2007
MAC 2312
S Hudson
1) (15 pts) Answer True or False in the margin. You don’t have to explain.
There is a point
c
in [0,
π
] such that sin(
c
) =
R
π
0
sin(
x
)
dx
.
If
f
is not bounded on [a,b] then it is not integrable on [a,b].
Displacement is always less than or equal to distance travelled.
For the shell method about the
y
axis, the formula is
V
=
R
b
a
2
πxf
(
x
)
dx
.
A cube is an example of a right cylinder.
2)(15pts) Find the area under
y
=
x
2
over the interval [0,1] using a limit of right end
point approximations. Draw a picture, show your work, and [hint!] use formulas like
Σ(
k/n
)
2
(1
/n
) and/or
n
(
n
+ 1)(2
n
+ 1)
/
6 along the way.
3) (10 points each) Calculate, and show work. Explain any nonobvious steps brieﬂy.
a)
R
π/
4
0
sin
2
x dx
=
b)
R
3
0

x

2

dx
=
c)
R
2
1
dx
√
5
x

1
=
d) Find the area between
y
=
x
2
and
x
=
y

2.
4)(15 pts) A ball is thrown upwards from ground level with an initial velocity of 16 ft/sec.
Find a formula for its height using antiderivative(s). [Recall that
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 Summer '08
 Storfer
 Calculus, Mean Value Theorem, dx, right endpoint approximations, fairly good averages

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