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Solutions to the calculus midterm i exam, focusing on integration and improper integrals. Topics covered include finding derivatives, evaluating integrals using substitution and l'hopital's rule, and calculating volumes of solids obtained by rotating regions about an axis.
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HONORS CALC II F 08, MIDTERM I
(1) Find the derivative of the function
∫ (^) x √x
et t dt. (Suggestion: Write the integral as a difference between two integrals, one from 0 to x and another from 0 to √x. Write f (t) = e tt. Set
I(x) =
∫ (^) x √x^ f^ (t)dt^ =^ I^1 (x)^ −^ I^2 (x),^ where^ I^1 (x) =
∫ (^) x
0
f (t)dt, I 2 (t) =
∫ √x
0
f (t)dt.
Hence by FTC and the chain rule,
I′(x) = f (x) − f^ (
x) 2
x
= e
x x
− e
√x
2 x
=^2 e
x (^) − e√x 2 x
(2) Evaluate the integral (^) ∫ dx x^2
x^2 − 16
Change x = 4 sec t. Then dx = 4 tan t sec tdt and x^2 − 16 = 16 tan^2 t. So
I =
tan t sec t 16 sec^2 t tan t
dt = 1 16
cos tdt = sin^ t 16
Now t = arccos(4/x), so sin t =
1 − cos^2 t, and we have
I =
1 − 16 /x^2 16
x^2 − 16 16 x
(3) Evaluate the integral (^) ∫ √ dx 1 + x +
x
(You don’t need to change variables, you can do algebraic manipulation using the identity (a + b)(a − b) = a^2 − b^2 ). Following the suggestion, we note that
√^1 1 + x +
x
1 + x +
x
1 + x −
√ x 1 + x −
x
1 + x −
x.
Therefore I =^2 3
(1 + x)^3 /^2 − x^3 /^2
(4) Evaluate the improper integral ∫ (^100)
0
t ln tdt.
Integrate by parts ∫ (^100)
0
t ln tdt =^2 3
t^3 /^2 ln t
100
0
0
t^1 /^2 dt =^2000 3
ln 100 − 4000 9
ln 10 − 1 3
1
Note that we have used L’Hospital’s rule to compute
lim t→ 0 t^3 /^2 ln t = lim t→ 0 ln^ t t−^3 /^2
= lim t→ 0
1 t − 32 t−^5 /^2
Alternatively, you can substitute u = ln
t = 12 ln t and then t = e^2 u, so dt = 2e^2 udu ∫ (^100)
0
t ln tdt =
∫ (^) ln 10
−∞
eu 2 u 2 e^2 udu = 4
∫ (^) ln 10
−∞
e^3 uudu =
3 e
3 u(u − 1 3 )
ln 10
−∞
3 (ln 10^ −^
(5) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis: y = 2, y = x^2 ; about the y axis. We’ll use y as the independent variable, and so
V = π
0
x^2 (y)dy = π
0
ydy = 2π
2