Integration
Calculus – Round 1
1. Write your 9 digit state FAMAT ID# in the appropriate spaces left justified, then bubble. Failure to do so may result in
disqualification. In the name blank, Print your name, in the subject blank print the test name, in the date blank print your
school name, no abbreviations.
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6. Any inappropriate behavior or any form of cheating by a participant will lead to the banning of the student and/or school
from future FAMAT competitions for up to one calendar year, at the discretion of the FAMAT Board.
7. If a student believes a test item is defective, select “NOTA” and file a resolution request explaining why.
8. In the case that a question has multiple correct answers, any of the answers will be counted as correct. DO NOT select
“NOTA.” This applies even if one of the correct answers is in more simplified form than another. If your answer is (is not one of
the choices), write a resolution request explaining why your answer should also be accepted.
9. If AN ANSWER CHOICE IS NOT COMPLETE, choose “NOTA.” For example, when solving a quadratic where there are two
solutions, a choice providing only one solution should NOT be chosen. On the other hand, if a question asks for "A solution…”,
a choice providing only one solution would be correct.
10. The phrase “two numbers” is to be interpreted as allowing the two numbers to be equal. The phrase “two distinct
numbers” means find two different numbers.
11. If a student files a resolution request claiming what the resolution center believes is a unique or highly unusual
interpretation of the problem, the resolution center may award that student credit while allowing the intended answer for all
other students.
12. Unless a question asks for an approximation or a rounded answer, give the EXACT answer.
13. The three-digit code for this test is:
48
D
49
C
50
B
Thank you to American Heritage Plantation for generously donating the printing of this test
2016 FAMAT Convention – Mu – Integration
Page 1 of 5
For all questions, choice (E) N.O.T.A. denotes that none of the aforementioned answers is correct.
For all questions, bxc denotes the floor function, which is equal to the greatest integer less than or
equal to x, and dxe denotes the ceiling function, which is equal to the least integer greater than or
equal to x. All functions are restricted to their standard domains unless otherwise stated.
Z 1 3
x2
√
1.
dx
3
x
0
6
13
A.
B. 1
C.
Z
4
3
D.
3
2
E. N.O.T.A.
5
2. Use Simpson’s Rule to approximate
(3x2 − 4x + 1) dx
1
A. 76
Z
3.
B. 80
C. 82
D. 110
E. N.O.T.A.
cot2 (θ) dθ
A. 13 cot3 (θ) + C
E. N.O.T.A.
4. Evaluate lim
n→∞
A. 0
B. − csc(θ) + C
n
X
C. − csc(θ) cot(θ) + C
D. − cot(θ) − θ + C
1
2
n + kn
π
B.
4
k=1
C. 1
D.
π
2
E. N.O.T.A.
Z x
2
2
e−t dt. Which of the following
5. The error function, erf(x), is defined as erf(x) = √
π 0
properly describes the nature of the error function on its domain?
A. Monotonically increasing, concave up
C. Monotonically decreasing, concave up
E. N.O.T.A.
Z
6.
0
A.
π
2
B. Monotonically increasing, concave down
D. Monotonically decreasing, concave down
dx
sin x + 1
1
2
B. ln 2
C. 1
D. Divergent
7. Find the volume of the solid formed when the region bounded by y =
x = 0, and x = 1 is revolved about the x-axis.
A. 1
B.
π
2
C. π
D. πe
E. N.O.T.A.
ex + e−x
ex − e−x
,y=
,
2
2
E. N.O.T.A.
2016 FAMAT Convention – Mu – Integration
8.
Page 2 of 5
sin5 (𝑥)
∫ cos2(𝑥) 𝑑𝑥 =
1
3
1
5
A. − 448 cos(7𝑥) + 320 cos(5𝑥) − 192 cos(3𝑥) − 64 cos(𝑥) + 𝐶
B.
1
sec 2 (𝑥)
2
1
− 4 cos(2x) + 2 ln|cos(𝑥)| + 𝐶
1
3
C. sec(x) − cos 3(𝑥) + 2 cos(x) + 𝐶
D. sin6(𝑥) sec(x) − cos 5(𝑥) +
E. N.O.T.A.
9.
10.
2 2𝑥 3 −3𝑥 2 +8𝑥+1
lim𝑛→∞ 𝑛 ∫1
3𝑥 3 +5𝑥 2 −7𝑥+𝑛+9
A. 0
B.
2
3
5 cos(x) + 𝐶
𝑑𝑥 =
C. 1
D. D.N.E.
E. N.O.T.A.
Find the area bounded by the 𝑥-axis and 𝑦 = sin⁡(𝑥) from 𝑥 = 0 to 𝑥 = 𝜋.
B. 1
A. 0
11.
10
cos 3(𝑥) −
3
1 𝑒 3𝑥 +1
∫0
A.
𝑒 𝑥 +1
1 2
𝑒
2
C. 2
D. 𝜋
E. N.O.T.A.
C. 𝑒 2 − 𝑒
D. 𝑒 2 − 𝑒 + 2
𝑑𝑥 =
1
−𝑒+2
B.
1 2
𝑒
2
3
−𝑒+2
1
E. N.O.T.A.
12.
4
5
5
If 𝑓(𝑥) is a continuous function with ∫1 𝑓(𝑥)𝑑𝑥 = 𝐴, ∫2 𝑓(𝑥)𝑑𝑥 = 𝐵, and ∫1 𝑓(𝑥)𝑑𝑥 = 𝐶,
4
which of the following is an expression for ∫2 𝑓(𝑥)𝑑𝑥?
A. 𝐴 + 𝐵 − 𝐶
E. N.O.T.A.
13.
∞
D. 𝐴 − 𝐵 + 𝐶
𝑥2
B.
𝜋
3
C.
𝜋
2
D. 𝜋
E. N.O.T.A.
Find the average value of 𝑓(𝑥) = 1 − 𝑥 4 on the interval 𝑥 ∈ [−1,1].
A.
15.
C. 𝐴 + 𝐵 + 𝐶
∫−∞ 𝑥 6+1 𝑑𝑥 =
A. 1
14.
B. 𝐶 − 𝐴 − 𝐵
5
8
B.
8
5
C.
4
5
1
2
C. 1
5
4
E. N.O.T.A.
D. 2
E. N.O.T.A.
D.
𝜋
∫0 sec(𝑥) 𝑑𝑥 =
A. 0
B.
2016 FAMAT Convention – Mu – Integration
Z
1
arcsin2 (x) dx
16.
0
π2
−2
4
A.
Z
Page 3 of 5
2
r
x+
17.
B.
π2
4
C.
π2
+1
4
D.
π2
+2
4
E. N.O.T.A.
q
√
x + x + . . . dx
0
A. − 76
∞
X
18. A =
n=2
B. 2
C.
19.
π
2
√
D.
14
3
E. N.O.T.A.
∞
X
1
1
and B =
ln (n n2 ). Use the integral test to determine which of the following
n ln n
n=1
is true.
A. Both A and B converge
C. A converges but B diverges
Z
19
6
B. Both A and B diverge
D. A diverges but B converges
E. N.O.T.A.
sec x − cos x dx
0
A. −2
B. −1
C. 1
D. 2
For questions 20 and 21, let R be the region bounded by y =
√
E. N.O.T.A.
π 2 − x2 and the x-axis.
20. Find the centroid of R.
B. 0, π3
A. (0, 1)
C. 0, 43
D. 0, π2
E. N.O.T.A.
21. Find the volume of the solid formed when R is rotated about the line y = − 34 x + 53 .
A.
π4
5
B.
π4
2
C. π 4
D.
4 4
π
3
E. N.O.T.A.
Z 1
π √
sec x
sin x ln x
dx and B =
dx.
A
+
B
is
equal
to
a
b
ln
+ c, where a, b,
π
π
x
cos2 x
4
4
4
√
and c are integers and a b is in simplest radical form. Find a + b + c.
Z
1
22. Let A =
A. 0
B. 1
C. 2
D. 3
E. N.O.T.A.
2016 FAMAT Convention – Mu – Integration
23.
3
∫0 √9 − 𝑥 2 𝑑𝑥 =
A. 0
24.
B. 9𝜋
1
C.
9𝜋
2
D. 𝜋
E. N.O.T.A.
D. 𝑒
E. N.O.T.A.
3
∫0 𝑥 2 𝑒 𝑥 𝑑𝑥 =
A.
25.
Page 4 of 5
𝑒−1
3
B.
𝑒
3
C. 1
For some tricky integrals, Leibniz devised a solution strategy known simply as “differentiation
∞ 𝑒 −𝛼𝑥 sin(𝑥)
under the integral sign.” Considering 𝐼(𝛼) = ∫0
∞ 𝑒 −𝑥 sin(𝑥)
∫0
A.
26.
𝑥
1
2
𝑥
𝑑𝑥, use this technique to solve
𝑑𝑥.
B.
𝜋
4
C. 1
D.
𝜋
2
E. N.O.T.A.
The Laplace transform is an operation that is particularly useful in physics and engineering. The
∞
Laplace transform of a function 𝑓(𝑡) is defined as ℒ{𝑓} = ∫0 𝑒 −𝑠𝑡 𝑓(𝑡)𝑑𝑡. Given this
information, find the Laplace transform of 𝑓(𝑡) = 𝑒 𝑎𝑡 for a constant 𝑎.
A.
27.
1
𝑠
B.
1
𝑠+𝑎
C.
1
𝑠−𝑎
D.
1
𝑎
E. N.O.T.A.
∫ arcsinh(𝑥) 𝑑𝑥 is equivalent to which of the following integrals?
A. ∫ 𝑥 sinh(𝑥) 𝑑𝑥
B. ∫ 𝑥 cosh(𝑥) 𝑑𝑥
1
C. ∫ sinh(𝑥) 𝑑𝑥
𝑥
1
D. ∫ cosh(𝑥) 𝑑𝑥
𝑥
E. N.O.T.A.
28.
Let {𝑥} be equal to the fractional part of a real number 𝑥, so {𝑥} = {
𝑥 − ⌊𝑥⌋, 𝑥 ≥ 0
.
𝑥 − ⌈𝑥⌉, 𝑥 < 0
𝜋
Given ∫−𝑒{𝑥}𝑑𝑥 = 𝐴𝜋 2 + 𝐵𝜋 + 𝐶𝑒 2 + 𝐷𝑒 + 𝐸, find 𝐴 + 𝐵 + 𝐶 + 𝐷 + 𝐸.
A. 0
B.
1
2
C. 2
D. 5
E. N.O.T.A.
2016 FAMAT Convention – Mu – Integration
Z
cot x sec2 x dx
A.
B.
C.
D.
E.
1 − cot x + C
1 + cot x + C
ln | sin x | + ln | cos x | + C
ln | csc(2x) + cot(2x) | + C
N.O.T.A.
29.
Z
Page 5 of 5
2
y dx
30.
0
A. 2
B. 2y − 1
C. 2y
D. 2xy
E. N.O.T.A.