Exam
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
1) Find the centroid of the plane region bounded by the curves.
x = 0, x = 3, y = 0, y = 4
1)
Find the center of mass of a thin plate of constant density covering the given region.
2) The region enclosed by the parabolas y = - x2 + 72 and y = x2
3) The region between the curve y = 4
x
2)
and the x-axis from x = 1 to x = 16
4) The region in the first and fourth quadrants enclosed by the curves y =
y = -4 and by the lines x = 0 and x =
1 + x2
3)
4
1 + x2
and
3
Find the center of mass of a thin plate covering the given region with the given density function.
5) The region bounded by x = y 2 and the line x = 25, with density !(x) = y 2
6) The region between the curve y = 3
function !(x) = 1
x
4)
and the x-axis from x = 1 to x = 9, with the density
5)
6)
x
7) The region between the curve y = 1 and the x-axis from x = 1 to x = 7 is revolved about
x
7)
the x-axis to generate a solid. Find the center of mass of a thin plate covering the region
if the plate's density at the point (x, y) is !(x) =
x.
The centroid of a triangle lies at the intersection of the triangle's medians, because it lies one-third of the way from
each side towards the opposite vertex. Use this result to find the centroid of the triangle whose vertices appear as
following.
8) (-10, 0), (10, 0), (0, 3)
8)
9) (0, 0), (7, 0), ( 7 , 10)
2
9)
10) (0, 0), (10, 0), (5, 7)
10)
1
Find the moment or center of mass of the wire, as indicated.
4 - x2 if the density
11)
12) Find the moment about the x-axis of a wire of constant density that lies along the curve
12)
13) Find the center of mass of a wire of constant density that lies along the first-quadrant
portion of the circle x2 + y 2 = 16.
13)
11) Find the center of mass of a wire that lies along the semicircle y =
of the wire is ! = 2sin ".
y=2
x from x = 0 to x = 3.
Solve the problem.
14) Find the centroid of the plane region bounded by the curves.
x = 2, x = 4, y = 3, y = 5
14)
15) Find the centroid of the plane region bounded by the curves.
x = -2, x = 4, y = -1, y = 5
15)
16) Find the centroid of the plane region bounded by the curves.
x = 0, y = 0, 2x + y = 2
16)
17) Find the centroid of the plane region bounded by the curves.
y = 0, y = 2x, x + y = 3
17)
18) Find the centroid of the plane region bounded by the curves.
y = x 2, y = 4
18)
19) Find the centroid of the plane region bounded by the curves.
y = x 2 - 2, y = -4, x = -3, x = 1
19)
20) Find the centroid of the plane region bounded by the curves.
y =x 2 , y = 8 - x 2
20)
21) Find the centroid of the plane region bounded by the curves.
y = 2x , y = 15 - x 2
21)
22) Find the centroid of the plane region (in the first quadrant) bounded by the curves.
y = x 3 ,x = y 3
22)
2
2
23) The region in the first quadrant bounded by the graphs of y = x and y = x is rotated
2
23)
around the line y = x. Find (a) the centroid of the region and (b) the volume of the solid
of revolution.
y
x
24) The region in the first quadrant bounded by the graphs of y = 2x and y = x 2 is rotated
around the line y = 2x. Find (a) the centroid of the region and (b) the volume of the solid
of revolution.
24)
y
x
25) The region in the first quadrant bounded by the graphs of y = 3x and y = x 3 is rotated
around the line y = 3x. Find (a) the centroid of the region and (b) the volume of the solid
of revolution.
y
x
3
25)
Answer Key
Testname: 150C06S04
1) ( 3 , 2)
2
24) (a) (1, 8 ); (b) 8 !
5
3
2) x = 0, y = 36
3) x = 7, y = 2 ln 16
3
25) (a) ( 8
15
4) x = 3 ln 4 , y = 0
2!
5) x = 125 , y = 0
7
6) x = 8 , y = 2
ln 9
ln 9
3/2 - 1
7) x = 7
,y=0
3( 7 - 1)
8) (0, 1)
9) ( 7 , 10 )
2 3
10) (5, 7 )
3
11) x = 0, y = 1 !
2
12) 28
3
13) x = 8 , y = 8
!
!
14) (3, 4)
15) (1, 2)
16) ( 1 , 2 )
3 3
17) ( 4 , 2 )
3 3
18) (0, 12 )
5
19) (- 21 , - 137 ) ≈ (-1.6154, -1.054)
13
130
20) (0, 4)
21) (-1, 22 )
5
22) ( 16 , 16 )
35 35
23) (a) (1, 4 ); (b) 4 !
5
3
2
4
3, 8
7
5
3 ) ≈ (0.9238, 1.9795); (b) 9 !
4
30