Calculus Area and Volume
FAMAT State Convention 2017
Choice E is NOTA, meaning ‘none of the above answers are correct’.
1. Rank the following integrals from smallest to largest:
1
I.
x
1
2
II.
dx
0
x
2
3
 x
III.
dx
B) III < I < II
E) NOTA
C) II < I < II
2. What is the area bound by y  tan( x) and the x-axis from x  0 to x 
1
A)  ln  2 
2
 1 dx
0
0
A) I < II < III
D) III < II < I
2
1
B)  ln
2
 2
C)  ln
 2
D) ln

4
?
 2
E) NOTA
3. Find the area bound by y  16   4 x  8 and the x-axis.
2
A) 2
B) 4
4. What is the average value of y  e
A)
e4 e2

2 6
B)
C) 8
x
e4 e2

2 6
D) 16
E) NOTA
D) 18e 4  6e 2
E) NOTA
on the interval  4,16 ?
C)
e4 e2

4 12
5. Larry has an equilateral triangle, A, with sides of length s. Inside triangle A, he inserts a smaller
equilateral triangle B, such that the vertices of B are touching the midpoints of the sides of A.
Inside triangle B, he inserts an even smaller equilateral triangle C, such that the vertices of C are
touching the midpoints of the sides of B. If he continues this pattern forever, with infinite
equilateral triangles, which of the following sums represents the combined area of all of the
triangles?
 3s 2 
A)  

n 1  4  4n  

 3s 2 

B)  
n
n 0  4  4  



 3  s 2 n 
C)  
  
n 0  4  4  



 3  s n 
D)  
  

n 0  4  4  

E) NOTA
6. Which of the following theorems require that a function, f ( x), is continuous on the closed
interval  a, b and differentiable on the open interval  a, b  ?
I. Rolle’s Theorem
II. The Mean Value Theorem for integrals
III. The Mean Value Theorem for derivatives
IV. The First Fundamental Theorem of Calculus
V. The Second Fundamental Theorem of Calculus
A) III, IV
B) II, IV, V
C) II, III, IV, V
D) I, III
E) NOTA
Page 1 of 5
Calculus Area and Volume
FAMAT State Convention 2017
7. Find the volume of the tetrahedron determined by the four following vectors:
 0, 0, 0 
 2, 0, 2 
 1, 3, 4 
 5, 6, 4 
A) 7
B) 11
C) 19
D) 48
E) NOTA
8. What is the volume obtained by rotating the region bound by the x-axis, y  x 2 and x  2
around the x-axis?
8
32
16
A)
B)
C)
D) 16
E) NOTA
3
5
5
9. The base of a solid is bound by y  3 x  1, x  0, x  8 and the x-axis. Find the volume of the
solid using semi-circular cross sections perpendicular to the y-axis.
16
128
64
64
A)
B)
C)
D)
E) NOTA
5
5
7
7
10. Where is the center of the tetrahedron determined by the vertices:
 0, 0, 0
 2, 0, 2
 1, 3, 4
 2

A)  0, , 2, 5 
 3

B)
3 9 5
C)  , , 
2 4 2
 6, 9, 8
5, 6, 4
4
5
D)  , 2, 
3
2
E) NOTA
11. If f ( x) is continuous and even on the interval  a, a , which of the following is true?
a
A)

a
a
B)

0
a
f ( x) dx  2  f ( x) dx
0
C)
a
f ( x) dx   f 1 ( x) dx
a
2

a
0
a
f ( x) dx    f ( x) dx
0
a
D)

a
2a
f ( x) dx 

f ( x) dx
E) NOTA
0
12. How many petals does the polar graph r  1  2cos  2  have?
A) 1
B) 2
C) 3
D) 4
E) NOTA
13. Consider a square inscribed within a circle with radius 5 centered at the origin. What is the
volume when this square is revolved around the line x  10 ?
A) 250
B) 500
C) 1000
D) 2000 2
E) NOTA
14. Consider a function f ( x)  ( x  1)2 . What is the area bound by f ( x) and f '( x) in quadrant I?
4
4
40
40
A)
B) 
C)
D) 
E) NOTA
3
3
3
3
Page 2 of 5
Calculus Area and Volume
FAMAT State Convention 2017
15. Compute:
n

   x  dx 

n 1  0

6
Where  x  represents the greatest integer that is less than (or equal to) x.
A) 13
B) 15
C) 35
D) 45
E) NOTA
16. Use a midpoint Riemann sum to approximate the area under the curve y    x  1  x  7 
2
using 3 subintervals of equal length from x  1 to x  7.
A) 168
B) 114
C) 96
D) 57
E) NOTA
1
17. Find the area of the largest rectangle that can be bound by the graph y  4  x 2 and the x-axis.
3
8
16
32
A) 2
B)
C)
D)
E) NOTA
3
3
3
18. What is the positive difference between the left and right hand Riemann Sum Approximations
 7 
for the area bound by y  2sin  2 x  and the x-axis on the interval 0,  using 7 subdivisions
 4 
of equal length?


A)
B) 1
C)
D) 2
E) NOTA
4
2
19. Find the area bound by the two curves f  x    x2  21 and g  x   x 2  3.
A) 0
B) 36
C) 72
D) 180
E) NOTA
20. Compute:
4
z x2

1 0
 xy  dy dx dz
1  z 
Be sure to integrate with respect to the correct variables. For example, if you are integrating with
respect to y, all other variables act as constants.
A) 0
B) 1
C) 35
D) 42
E) NOTA
21. Patrick, who is 5 feet tall, is walking away from a 12 foot tall lamp post at exactly 3 feet /
second. When he is 27 feet away from the lamp, how fast is his shadow increasing in length (in
ft/s)?
15
1
1
36
A)
B)
C)
D)
E) NOTA
7
4
9
7
Page 3 of 5
Calculus Area and Volume
FAMAT State Convention 2017
22. This famous mathematician aided in proving that Mars’ orbit was not a perfect circle, but
elliptical. His early theory of triangulation was too inaccurate to be considered correct. But later,
using the distances between the planets and the sun at various time periods, he contributed to
the discovery of the area of an ellipse (an important conic application). He concludes his finding:
“…Clearly, then, [what is to be said] is this: the orbit of the planet [Mars] is not a circle, but
comes in gradually on both sides and returns again to the circle’s distance at perigee. They are
accustomed to call the shape of this sort of path an oval.”
A) Isaac Newton
D) Johannes Kepler
B) Archimedes of Syracuse
E) NOTA
C) Joseph-Louis Lagrange
23. Squidward, for his sculpture, needs to find the volume of the largest cylinder that can fit inside
of a right circular cone of height 10 ft and base radius 3 ft. Help him out.
A) 2 ft 3
10  5 3
2
24. Compute:

5
2
A)
10
 ft 3
3
B)
x4
10 x  x 2
10  5 3 

2
6
C)
20 2 3
 ft
3
D)
40
 ft 3
3
E) NOTA
C)
55 3 

2
6
D)
5 3 5 

2
6
E) NOTA
dx
B)
5 

2 6
25. Find the area of an ellipse with major axis length 8 and minor axis length 2.
B) 4
A) 2
C) 8
D) 16
E) NOTA
26. Help Spongebob find the area of the region bound by the x-axis, y-axis, f ( x) 
inverse f 1 ( x) 
A) 0
5
10
x .
3
3
20
B)
3
C) 10
D)
40
3
27. Help Sandy find the total area of the region bound by the y-axis, f ( x) 
f 1 ( x) 
A) 0
3
x  2 and it’s
5
E) NOTA
3
x  2 and
5
5
10
x .
3
3
B)
20
3
C) 10
D)
40
3
E) NOTA
Page 4 of 5
Calculus Area and Volume
FAMAT State Convention 2017
28. SpaceX is a private aerospace corporation experimenting with reusing rockets that they
previously send into space. To reuse the rocket, they must land it on the ground in a very
specific spot. For example, they need to land a rocket on a pad determined by the polar curve
r  2  3sin( ), but a portion of the pad is obstructed by debris, so they cannot land in the area
represented by r  2. Given that the rocket is functioning perfectly, find the probability SpaceX
can make a successful landing on the remaining portion of the landing pad.
A)
9
17
B)
48  9
34
C)
25
68
D)
32  9
4
E) NOTA
29. Compute:
   xx   x 
1  ln  x   x  dx
3
A) 364
B) 27ln  3
C) 26
D) ln  3e 
E) NOTA
30. The amount of secrets in Patrick’s secret box is equal to the surface area of the polar curve

r  sec   when it is revolved around the y-axis for 0    . How many secrets are in his
6
secret box?
A)
2 3
3
B) 2
C)
3 6
2
D) 4
E) NOTA
Page 5 of 5