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Volume: Shell Method Worksheet

Volume: Shell Method Worksheet
  • Page 1
 1.  
Find the volume of the solid of revolution formed by revolving the region bounded by f(x) = x2 and the x-axis (0 ≤ x ≤ 2) about y-axis. [Use shell method.]
a.
4π cubic units
b.
32π cubic units
c.
8π cubic units
d.
12π cubic units
e.
16π cubic units


Solution:

Equation of the curve is f(x) = x2, (0 ≤ x ≤ 2)

In the shell method, the volume of solid of revolution formed by revolving the region bounded by y = f(x) and the x - axis (axb), about y - axis is given by, V = 2π ab x · f(x) dx

Volume of solid of revolution, V = 2π 02 x (x2) dx

= 2π 02 x3 dx

= 2π [x44]02

= 2π (16 / 4) = 8π


Correct answer : (3)
 2.  
Find the volume of the solid of revolution formed by revolving the region bounded by f(y) = y - y2 and the y-axis (0 ≤ y ≤ 1) about x-axis. [Use shell method.]
a.
π6 cubic units
b.
7π12 cubic units
c.
π12 cubic units
d.
π3 cubic units
e.
14π12 cubic units


Solution:

Equation of the curve is f(y) = y - y2, (0 ≤ y ≤ 1)

In the shell method, the volume of solid of revolution formed by revolving the region bounded by x = f(y) and the y - axis (ayb), about x - axis is given by, V = 2π ab y f(y) dy

Volume of solid of revolution, V = 2π 01 y (y - y2) dy

= 2π 01 (y2 - y3) dy

= 2π [y33-y44]01

= 2π [1 / 3 - 1 / 4]

= 2π · 1 / 12 = π6 cubic units


Correct answer : (1)
 3.  
Find the volume of the solid of revolution formed by revolving the region bounded by f(x) = ex, x-axis, x = 0 and x = 2 about y-axis. [Use shell method.]
a.
2π (1 - e2) cubic units
b.
π2 (1 - e2) cubic units
c.
π (1 - e2) cubic units
d.
π (1 + e2) cubic units
e.
2π (1 + e2) cubic units


Solution:

Equation of the curve is f(x) = ex, (0 ≤ x ≤ 2)

In shell method, the volume of solid of revolution formed by revolving the region bounded by y = f(x) and the x - axis (axb), about y - axis is given by, V = 2π ab x f(x) dx

Volume of the solid of revolution = 2π 02 x · ex dx

= 2π [xex - ex]02

= 2π [(2e2 - e2) - (0 - 1)]

= 2π [(e2) + 1]

= 2π (1 + e2) cubic units


Correct answer : (5)
 4.  
Find the volume of the solid of revolution formed by revolving the region bounded by f(x) = sin x, x-axis, x = 0 and x = π2 about y-axis. [Use shell method.]
a.
2π cubic units
b.
4π cubic units
c.
π2 cubic units
d.
π4 cubic units
e.
π cubic units


Solution:

Equation of the curve is f(x) = sin x, (0 ≤ xπ2)

In the shell method, the volume of solid of revolution is formed by revolving the region bounded by y = f(x) and the x - axis (axb), about y - axis is given by, V = 2π ab x · f(x) dx

Volume of solid of revolution, V = 2π 0π/2 x sin(x) dx

= 2π [sin x - x cos x]0π/2

= 2π(1) = 2π


Correct answer : (1)
 5.  
Find the volume of the solid of revolution formed by revolving the region bounded by f(y) = y and the y-axis (0 ≤ y ≤ 2) about x-axis. [Use shell method.]
a.
16π5 cubic units
b.
162π5 cubic units
c.
82π5 cubic units
d.
32π5 cubic units
e.
8π5 cubic units


Solution:

Equation of the curve is f(y) = y, (0 ≤ y ≤ 2)

In the shell method, the volume of solid of revolution formed by revolving the region bounded by x = f(y) and y - axis (ayb), about x - axis is given by, V = 2π ab y f(y) dy

Volume of solid of revolution, V = 2π 02 yy dy

= 2π 02 y32 dy

= 2π [y5252]02

= 4π5(42)

= 162π5


Correct answer : (2)
 6.  
Find the volume of the solid of revolution formed by revolving the region bounded by f(x) = 1 + 4x and the x-axis (2 ≤ x ≤ 4) about y-axis. [Use shell method.]
a.
28π
b.
48π
c.
56π
d.
68π
e.
14π


Solution:

Equation of the curve is f(x) = 1 + 4x, (2 ≤ x ≤ 4)

In the shell method, the volume of solid of revolution formed by revolving the region bounded by y = f(x) and the x - axis (axb), about y - axis is given by, V = 2π ab x f(x) dx

Volume of solid of revolution, V = 2π 24 x (1 + 4x) dx

= 2π 24 (x + 4) dx

= 2π [x22 + 4x]24

= 2π (14)

= 28π


Correct answer : (1)
 7.  
Find the volume of the solid of revolution formed by revolving the region bounded by f(x) = ln x, x-axis, x = 1 and x = e about y-axis. [Use shell method.]
a.
2π (e2 - 1) cubic units
b.
π (e2 - 1) cubic units
c.
2π (e2 + 1) cubic units
d.
π2(e2+1) cubic units
e.
π2(e2-1) cubic units


Solution:

Equation of the curve is f(x) = ln x, (1 ≤ xe)

In the shell method, the volume of solid of revolution formed by revolving to region bounded by y = f(x) and the x - axis (axb), about y - axis is given by, V = 2π ab x f(x) dx

Volume of solid of revolution, V = 2π 1e x ln x dx

= 2π [x22 (ln x - 1 / 2)]1e

= 2π [e22(1 - 1 / 2) - 1 / 2 (- 1 / 2)]

= 2π (e24 + 1 / 4)

= π2(e2+1) cubic units


Correct answer : (4)
 8.  
Find the volume of the solid of revolution formed by revolving the region bounded by f(y) = y2 + y + 1, y-axis, y = 1 and y = 2 about x-axis. [Use shell method.]
a.
91π6 cubic units
b.
41π3 cubic units
c.
23π3 cubic units
d.
35π3 cubic units
e.
33π3 cubic units


Solution:

Equation of the curve is f(y) = 1 + y + y2, (1 ≤ y ≤ 2)

In the shell method, the volume of solid of revolution formed by revolving the region bounded by x = f(y) and the y - axis (axb), about x - axis is given by, V = 2π ab y f(y) dy

Volume of solid of revolution, V = 2π 12 y(1 + y + y2) dy

= 2π [y22+y33 + y44]12

= 2π (2 + 8 / 3 + 4 - 1 / 2 - 1 / 3 - 1 / 4)

= 2π (6 + 7 / 3- 3 / 4) = 91π6 cubic units.


Correct answer : (1)
 9.  
Find the volume of solid formed by revolving the region bounded by y = 22x and y = x2 about x-axis. [Use Washer method.]
a.
24π5 cubic units
b.
56π5 cubic units
c.
112π5 cubic units
d.
48π5 cubic units
e.
32π5 cubic units


Solution:


Equations of curves are y = 22x and y = x2

These two curves intersect at (0, 0) and (2, 4). So, x varies from 0 to 2 in the region bounded by y = 22x and y = x2.

In Washer method, the volume of solid formed by revolving the region bounded by y = f(x) and y = g(x) is given by, V = πab [(f(x))2 - (g(x))2] dx, where f(x) is the top curve and g(x) is the bottom curve.

Volume of solid of revolution, V = π0 2 [(22x)2 - (x2)2] dx

= π02 (8x - x4) dx

= π [4x2 - x55]02 = π (16 - 32 / 5) = 48π5 cubic units.


Correct answer : (4)
 10.  
Find the volume of solid formed by revolving the region bounded by y = 0 and y = x2 - 4 about x-axis. [Use Washer method.]
a.
512π15 cubic units
b.
47π cubic units
c.
51π cubic units
d.
17π cubic units
e.
85π cubic units


Solution:


Equations of curves are y = 0 and y = x2 - 4

These two curves intersect at (- 2, 0) and (2, 0). So, x varies from - 2 to 2 in the region bounded by y = 0 and y = x2 - 4.

In Washer method, the volume of solid formed by revolving the region bounded by y = f(x) and y = g(x) is given by, V = πab [(f(x))2 - (g(x))2)] dx, where f(x) is the top curve and g(x) is bottom curve.

Volume of solid of revolution, V = π-22 [(x2 - 4)2 - (0)2] dx

= 2π 02 [16 + x4 - 8x2] dx

= 2π [16x + x55-8x33]02

= 2π (32 + 32 / 5 - 64 / 3) = 512π15 cubic units.


Correct answer : (1)
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