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Directions:
1. There are 14 questions worth a total of 60 points.
2. Questions 1 - 10 are Multiple Choice worth 4 points each to be answered on
the supplied SCANTRONS.
3. Questions 11 - 14 are Free Responses worth 5 points each and are to be
answered in the space provided on the test.
4. Read all the questions carefully.
5. For the Free Response, you must show all work in order to receive credit!!
6. When possible, box your answer, which must be complete, organized, and
exact unless otherwise directed.
7. Always indicate how a calculator was used (i.e. sketch graph, etc. …).
8. No calculators with QWERTY keyboards or ones like TI-89 or TI-92 that
do symbolic algebra may be used.
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Signature Date
MAT 266
TEST 2 - ANSWERS
SoMSS, ASU
- Find the area of the region bounded by the curves y = x^2^ − 2 x & y = x + 4. Select the correct answer. a.
b. 25 3 c. 20 d.
e. None of these
- Find the volume of the solid obtained by rotating the region bounded by y = x^2 and x = y^2 about the x -axis. Select the correct answer. a. 5
6 π (^) b. 10
3 π (^) c. 5
(^2) d. 5
π (^) e. None of these
- Which of the following integrals is equal to 1.25? Select the correct answer.
a.
1
0
(^1) dx ∫ (^) x b.
1
0
(^1) dx ∫ (^) x c.
1
0
(^1) dx ∫ (^) x d.
1 2 0
(^1) dx ∫ (^) x e. none of these
- A spring has a natural length of 22 cm. If a force of 15 N is required to keep it stretched to a length of 32 cm, how much work is required to stretch it from 22 cm to 40 cm? Select the correct answer. a. 3.43 J b. 1.93 J c. 2.43 J d. 3.93 J e. None of these
- Find lim(3 n →∞ + e −^2 n ) Select the correct answer a. 3 b. 0 c. 4 d. 2 e. None of these
- A rope, 40 ft long, weighs 0.8 lb/ft and hangs over the edge of a building 110 ft high. How much work is done in pulling the rope to the top of the building? Select the correct answer. a. 640 ft-lb b. 590 ft-lb c. 641 ft-lb d. 489 ft-lb e. None of these
- Find the sum of the series 4
n
n
∞
∑ Select the correct answer
a.
b.
c.
d.
e. None of these
Answer: d
Answer: b
Answer: c
Answer: c
Answer: a
Answer: a
Answer: b
- The tank shown is full of water. Given that water weighs work required to pump the water out of the tank.
Solution: Volume of the ith^ element =
Weight of the ith^ element = 62.5 (25 )
Work done to pump out the ith
Work done to pump out all the water
- Evaluate the integral or
Solution:
First observe
∫ (^) x (ln x ) ∫
So,
2 2 = lim[ ]
= lim[ ]
x x x x
∞ ∫ ∫
FREE RESPONSE
The tank shown is full of water. Given that water weighs 62.5 lb/ft^3 work required to pump the water out of the tank.
element = π (25 − xi^2 )∆ x 62.5 (25 2 ) π − xi ∆ x element = 62.5 π (25 − xi^2 ) xi ∆ x
Work done to pump out all the water = ith^ element =
5 2 0
∫^ 62.5^ π^ (25^ −^ x^ )^ xdx^ =^ 9765.625π ft^ − lb
or show that it is divergent: (^4) 2
(ln )
dx x x
∞ ∫.
4 4
3
(^1) ( ln ) (ln ) 1 3[ln ]
dx u du u x x x
x
∫ ∫
4 4 2 2
3 2
3 3 3
(^1) lim 1 (ln ) (ln ) 1 = lim[ ] 3(ln )
= lim[ 1 1 ]^1 3(ln ) 3(ln 2) 3(ln 2)
t t x t t x
t
dx dx x x x x
x
t
→∞
→∞ =
→∞
∫ ∫
62.5 lb/ft 3 and R = 5 ft, find the
62.5 π (25 − x ) xdx = 9765.625π ft − lb
- Set up the integral for the volume of the solid obtained by rotating the region bounded by y = x^3 and x = y^3 in the first quadrant, about the line x =− 1. ( Do not evaluate the integral )
Solution: Outer radius of the cross-section = y 1/3^ + 1 Inner radius of the cross-section = y^3 + 1
Volume =
1 1/3 2 3 2 0
π (^) ∫ [( y + 1) − ( y +1) ] dy
- The height of a monument is 20 m. A horizontal cross-section at a distance x meters from the top is an equilateral triangle with side x /4 meters. Find the volume of the monument.
Solution:
The height of an equilateral triangle with side-length equal to s units =
s
So, area of the equilateral triangle = 1 3 3 2 2 2 4
⋅ s ⋅ s = s
For our problem, area of the equilateral triangle = 3 2 64
x
Hence, volume of the monument =
20 2 0
∫^ x^ dx^ =
